7. Commutative algebra. Algebraic number theory.

"Abstract" commutative algebra is a recent creation, but its development can not be understood except in relation to that of algebraic number theory and algebraic geometry which gave it birth.

It was possible to conjecture without too much improbability that the famous "proof" that Fermat claimed to have of the impossibility of the equation $x^p + y^p = z^p$ for $p$ an odd prime number and $x, y, z$ integers $\neq 0$, was based on the decomposition

in the ring $\mathbf{Z}[\zeta]$ (where $\zeta \neq 1$ is a $p$th root of unity), and on an argument about divisibility in this ring, supposing it to be principal. It is found in any case that an analogous argument is outlined by Lagrange ([191], v. II, p. 531); it is by arguments of this kind, with different variants (notably alterations of the variables with the purpose of lowering the degree of the equation) that Euler ([108 a], v. I, p. 488)$^1$ and Gauss ([124 a], v. II, p.387) prove Fermat's theorem for $p = 3$, Gauss (loc. cit. and Dirichlet ([92], v. I, p. 42) for $p = 5$, and Dirichlet the impossibility of the equation $x^{14} + y^{14} = z^{14}$ ([92], v. I, p. 190). Finally, in his first research on the theory of numbers, Kummer had believed that he had obtained in this way a general proof, and it is no doubt this error (which was pointed out to him by Dirichlet) that led him to his studies on the arithmetic of cyclotomic fields, from where he was at last to succeed in deducing a correct version of his proof for prime numbers $p < 100$ [188 d].

From another angle, the famous memoir of Gauss of 1831 on biquadratic residues, whose results are deduced from a detailed study of divisibility in the ring $\mathbf{Z}[i]$ of "Gaussian integers" ([124 a], v. II, p. 109) showed what advantage there could be for the classical problems of number theory by the extension of the notion of divisibility to algebraic numbers;$^2$ so it is not surprising that between 1830 and 1850 this theory was the object of the work of many

$^1$ In his proof, Euler proceeds as if $\mathbf{Z}[\sqrt{-3}]$ were a principal ideal ring, which is not the case; however his argument can be made correct by considering the leader of $\mathbf{Z}[\rho]$ ($\rho$ a cube root of unity) on $\mathbf{Z}[\sqrt{-3}]$ (cf. [288], p. 190).

$^2$ The research of Gauss on division of the lemniscate and the elliptic functions linked to this curve, unpublished during his lifetime, but dating from around 1800, must have brought him already at that time to reflect on the arithmetical properties

German mathematicians, Jacobi, Dirichlet and Eisenstein first, then, a little later, Kummer and his pupil and friend Kronecker. We should not speak here of the theory of units, too restricted to number theory, where progress was rapid, Eisenstein obtaining the structure of the groups of units for the cubic fields, Kronecker for the cyclotomic fields, shortly before Dirichlet, in 1846 ([92], v. I, p. 640) proves the general theorem, which Hermite had almost reached on his own account ([159], v. I, p. 159). Much more difficult would appear to be the question (central to the whole theory) of the decomposition into prime factors. Since Lagrange had given examples of numbers of the form $x^2 + Dy^2$ ($x, y, D$ integers) having divisors that are not of the form $m^2 + Dn^2$ ([191], v. II, p. 465), it was known substantially that it was not to be expected that in general the rings $\mathbf{Z}[\sqrt{-D}]$ should be principal ideal rings, and to the temerity of Euler a great circumspection had succeeded; when Dirichlet, for example, proves that the relation $p^2 - 5q^2 = r^5$ ($p, q, r$ integers) is equivalent to $p + q\sqrt{5} = (x + y\sqrt{5})^5$ for $x, y$ integers, he limits himself to indicating in a note that "there are analogous theorems for many other prime numbers [other than 5]" ([92], v. I, p. 31). With the memoir of Gauss of 1831 and the work of Eisenstein on cubic residues [102 a], one had, it is true, some advanced studies of the arithmetic of the principal ideal rings $\mathbf{Z}[i]$ and $\mathbf{Z}[\rho]$ ($\rho = (-1 + i\sqrt{3})/2$, a cube root of unity) in perfect analogy with the theory of rational integers, and with these examples at least, the tight link between arithmetic in quadratic fields and the theory of binary quadratic forms developed by Gauss was very apparent; but for the general case a "dictionary" was missing which would have allowed the treatment of a quadratic field by simple translation of the theory of Gauss.$^3$

In fact, it is not for quadratic fields, but indeed for cyclotomic fields (and for reasons that would not appear clearly until much later (cf. p. 98)) that the enigma would first be resolved. From 1837, Kummer, an analyst to begin with, turns to the arithmetic of cyclotomic fields, which is not going to cease from keeping him busy almost exclusively for 25 years. Like his predecessors, he studies divisibility in the rings $\mathbf{Z}[\zeta]$, where $\zeta$ is a $p$th root of unity $\neq 1$ ($p$ an odd prime); he soon sees that, there also, rings that are not principal ideal rings occur, blocking all progress in extending the laws of arithmetic [188 a], and it is only in 1845, at the end of 8 years of effort, that the light finally dawns, thanks to his definition of "ideal numbers" [188 c and d].

What Kummer is doing reduces, in modern language, to defining valuations on a field $\mathbf{Q}[\zeta]$: they are in bijective correspondence with his "prime ideal numbers", the "exponent" with which such a factor appears in the "decomposition" of a number $x \in \mathbf{Z}[\zeta]$ being none other than the value of $x$ in

$^3$ of the ring $\mathbf{Z}[i]$, division by numbers of this ring playing a very important role in the theory; see what Jacobi says on this point ([171], v. VI, p. 275) as well as the calculations referring to these questions found in the papers of Gauss ([124 a], v. II, p. 411; see also [124 a], v. X_2, pp. 33 ff.).

the corresponding valuation. As the conjugates of $x$ also belong to $\mathbf{Z}[\zeta]$, and that their product $N(x)$ (the "norm" of $x^4$) is a rational integer, the "prime ideal factors" to be defined had also to be "factors" of rational prime numbers, and to give it its definition, one could limit oneself to saying that they were the "prime ideal divisors" of a prime number $q \in \mathbf{Z}$. For $q = p$, Kummer had already substantially proved [188 a] that the principal ideal $(1 - \zeta)$ was prime and that its $(p - 1)$th power was the principal ideal $(p)$; this case did not therefore raise any new problems. For $q \neq p$, the idea that seems to have guided Kummer is to replace the cyclotomic equation $\Phi_p(z) = 0$ by the congruence $\Phi_p(u) \equiv 0 \pmod{q}$, in other words to decompose the cyclotomic polynomial $\Phi_p(X)$ over the field $\mathbf{F}_q$, and to associate with each irreducible factor of this polynomial a "prime ideal factor". A simple case (explicitly quoted in the Note [188 b] where Kummer announces his results without proof) is that where $q \equiv 1 \pmod{p}$; if $q = mp + 1$ and if $\gamma \in \mathbf{F}_q$ is a $(q - 1)$st primitive root of 1, we have, in $\mathbf{F}_q[X]$,

since $\gamma^{pm} = 1$. Associating then with each factor $X - \gamma^{km}$ a "prime ideal factor" $q_k$ of $q$, Kummer says that an element $x \in \mathbf{Z}[\zeta]$, of which $P$ is the minimal polynomial over $\mathbf{Q}$, is divisible by $q_k$ if in $\mathbf{F}_q$ we have $P(\gamma^{km}) = 0$; to sum up, in modern language, he writes the quotient ring $\mathbf{Z}[\zeta]/q \mathbf{Z}[\zeta]$ as a direct product of fields isomorphic to $\mathbf{F}_q$. For $q \not\equiv 1 \pmod{p}$, the irreducible factors of $\Phi_p(X)$ in $\mathbf{F}_q[X]$ are no longer of the first degree, and it will therefore be necessary to substitute for $X$ in $P(X)$ "Galois imaginary" roots of the factors of $\Phi_p$ in $\mathbf{F}_q[X]$. Kummer avoids this difficulty by passing on, as we would say today, to the decomposition field $K$ of $q$: if $f$ is the smallest integer such that $q^f \equiv 1 \pmod{p}$, and if one puts $p - 1 = ef$, $K$ is none other than the subfield of $\mathbf{Q}(\zeta)$ constructed from the invariants of the subgroup of order $f$ of the Galois group (cyclic of order $p - 1$) of $\mathbf{Q}(\zeta)$ over $\mathbf{Q}$; in other words it is the unique subfield of $\mathbf{Q}(\zeta)$ which is of degree $e$ over $\mathbf{Q}$; it was very well known since the Disquisitiones of Gauss, being generated by the "periods"

4 The notion of norm of an algebraic number goes back to Lagrange: if $\alpha_i$ ($1 \leq i \leq n$) are the roots of a polynomial of degree $n$, he even considers the "norm form" $N(x_0, x_1, \ldots, x_{n-1}) = \prod_{i=1}^n (x_0 + \alpha_i x_1 + \ldots + \alpha_i^{n-1} x_{n-1})$ in the variables $x_i$, which had no doubt been suggested to him by his research on the solution of equations and the "Lagrange resolvants" ([191], v. VII, p. 170). It is to be noticed that it is the multiplicative property of the norm that leads Lagrange to his identity on binary quadratic forms, from where Gauss was to extract the "composition" of these forms ([124 a], v. II, p. 522). On the other hand, when the theory of algebraic numbers starts up around 1830, it is very often under the form of the solution of equations $N(x_0, \ldots, x_{n-1}) = \lambda$ (in particular with $\lambda = 1$ in order to seek units) or the study of "norm forms" (also called "decomposable forms") that the problems are set out; and even in recent work, the properties of these particular diophantine equations are used fruitfully, notably in the theory of $p$-adic numbers (Skolem, Chabauty).

We will not take up the history of the subsequent work of Kummer on cyclotomic fields, as far as the determination of the number of classes and the application of the proof of Fermat's theorem in various cases is concerned. Let us only mention the way in which, in 1859, he extends his method in order to obtain (at least partially) the "prime ideal numbers" in a "Kummer field" $\mathbf{Q}(\zeta, \mu)$, where $\mu$ is a root of an irreducible polynomial $P(X) = X^p - \alpha$, with $\alpha \in \mathbf{Z}[\zeta]$ [188 e]. It is interesting that Kummer sees this problem precisely as considering $\mathbf{Q}(\zeta, \mu)$ as a cyclic extension of the field $\mathbf{Q}(\zeta)$ taken as "base field":$^6$ he starts with a "prime ideal number" $q$ of $\mathbf{Z}[\zeta]$, that he assumes divides neither $p$ nor $\alpha$, and this time, he examines (in modern terms) the

$^5$ He only takes up there an argument from Kronecker's dissertation, referring to classes of solutions of equations of the form $N(x_0, x_1, \ldots, x_{n-1}) = a$ ([188 a], v. II, p. 25). On the other hand Kummer often alludes to results that Dirichlet would have obtained about equations of this type (for an arbitrary field of algebraic numbers); but these results have not been published, nor found in the papers of Dirichlet.

$^6$ In his memoir on quadratic forms with coefficients in the ring of Gaussian integers ([92], v. I, pp. 533-618), Dirichlet had, in various places, been led to consider

7. COMMUTATIVE ALGEBRA. ALGEBRAIC NUMBER THEORY.

polynomial $\overline{P}(X) = X^p - \overline{\alpha}$ in the residual field $k$ of the valuation of $\mathbf{Q}(\zeta)$ corresponding to $q$ ($\overline{\alpha}$ being the canonical image of $\alpha$ in $k$). As $\mathbf{Q}(\zeta)$ is the field of $p$th roots of unity, $\overline{P}$ is, either irreducible over $k$, or a product of factors of the first degree. In the first case, Kummer says that $q$ remains prime in $\mathbf{Z}[\zeta, \mu]$; in the second, he introduces elements $w_i (1 \leq i \leq p)$ of $\mathbf{Z}[\zeta]$ whose images in $k$ are the roots of $\overline{P}$, and he associates with each index $i$ a prime factor ideal $\tau_i$ of $q$; setting next $W_i(X) = \prod_{j \neq i} (X - w_j)$, he says that, for a polynomial $f$ with coefficients in $\mathbf{Z}[\zeta]$, $f(\mu)$ contains the factor ideal $\tau_i$ $m$ times if one has

but

To sum up, he obtains in this way the valuations of $\mathbf{Q}(\zeta, \mu)$ unramified over $\mathbf{Q}$, which is sufficient for the applications which he has in mind.

Kummer had the luck to meet, in his study of the particular fields to which his research on Fermat’s theorem had first led him, a number of fortuitous circumstances which rendered the study much more feasible. The extension of Kummer’s results to the general case presented formidable difficulties and was to cost years of efforts.

After Kronecker and Dedekind, who hold the lead roles in it, the history of the theory of algebraic numbers, during the 40 years that follow the discovery of Kummer, is not without a reminder (but happily without the same acrimonious character) of the rivalry of Newton and Leibniz 180 years earlier, over the invention of the infinitesimal Calculus. Student and soon a colleague of Kummer at Berlin, Kronecker (whose thesis, as we have seen, was useful to Kummer at an essential point of his theory) was very closely interested in “ideal numbers” with the purpose of applying them to his own research; and we admire his astonishing insight when we see him, already in 1853 ([186 a], v. IV, p. 10), stating the general theorem on the structure of abelian extensions of $\mathbf{Q}$ and, what is perhaps even more remarkable, creating, in the following years, the theory of complex multiplication and discovering the first shoots of the theory of class fields ([186 a], v. IV, pp. 177-183 and 207-217). A letter from Kronecker to Dirichlet; in 1857 ([186 a], v. V, pp. 418-421), shows him already at that time, in possession of a generalisation of Kummer’s theory, which besides is confirmed by Kummer himself in one of

the relative norm of the field $\mathbf{Q}(\sqrt{D}, i)$ over a quadratic subfield $\mathbf{Q}[\sqrt{D}]$. In the same way, Eisenstein, studying the 8th roots of unity, considers the field that they generate as a quadratic extension of $\mathbf{Q}(i)$ and uses the norm relative to this sub field ([102 b], p. 253). But Kummer’s work is the first example of a deep arithmetical study of a “relative field”.

his own works ([188 e], p. 57), and Kronecker will often refer to this theory in his memoirs between 1860 and 1880.7

But although at this time none of the mathematicians of the German school of number Theory were ignorant of this work of Kronecker, this latter seems only to have communicated the principles of his methods to a restricted circle of friends and pupils, and when he finally decides to publish them, in his memoir of 1881 on the discriminant ([186 a], v. II, pp. 193-236), and especially in his great "Festschrift" of 1882 ([186 a], v. II, pp. 237-287), Dedekind can not prevent himself from expressing his surprise ([79], v. III, p. 427), having imagined quite different procedures from the echoes which he had heard ([79], v. III, p. 287). Besides Kronecker was far from having to the same degree the remarkable gifts of exposition and clarity of Dedekind, and it is therefore not surprising that it is mostly the methods of this latter, published already in 1871, which formed the framework of algebraic number theory; as interesting as it is, the method of "adjunction of indeterminates" of Kronecker, as far as Number Theory is concerned, is no more to our eyes than a variant of that of Dedekind, and it is mainly in another direction, aimed towards Algebraic Geometry, that Kronecker's ideas acquired all their importance for the history of Commutative Algebra, as we will see later.

For reasons that could only appear clearly much later, a first preamble to any attempt at a general theory was of course the clarification of the notion of algebraic integer. This was acquired in about 1845-1850, although it is fairly difficult to date its apparition in a precise way; it appears likely that it is the idea of a system stable under addition and multiplication (or, more precisely, what we now call a $\mathbf{Z}$-algebra of finite rank) that, more or less consciously, had led to the general definition of algebraic integers: this definition is inevitable when one forces a $\mathbf{Z}$-algebra of the type $\mathbf{Z}[\theta]$ to be of finite rank, by analogy with the ring $\mathbf{Z}[\zeta]$ generated by a root of unity, which was at the centre of the preoccupations of the arithmeticians of the time. So it is that when, independently, Dirichlet ([92], v. I, p. 640), Hermite ([159], v. I, pp. 115 and 146) and Eisenstein ([102 c], p. 236) introduce the notion of algebraic integer, they do not seem to consider that it consists of a new idea nor to believe that it is useful to make a detailed study thereof; only Eisenstein proves substantially (loc. cit.) that the sum and the product of two algebraic integers are algebraic integers, besides without claiming that this result is original.

A point which was much better hidden was the determination of the rings in which one could hope to generalise Kummer's theory. This latter, in his first note [188 b], does not hesitate to affirm that he can find by his methods the theory of binary quadratic forms of Gauss by considering the rings $\mathbf{Z}[\sqrt{D}]$ ($D$ an integer); he never developed this idea, but it certainly appears that neither he nor anyone before Dedekind was able to see that the

7 On the evolution of his ideas on this subject, see the very interesting introduction to his memoir of 1881 on the discriminant ([186 a], v. II, p. 195).

unique decomposition into prime “ideal” factors is not possible in rings of the form $\mathbf{Z}[\sqrt{D}]$ when $D \equiv 1 \pmod{4}$ (although the example of the cube roots of unity showed that the ring $\mathbf{Z}[\rho]$ considered since Gauss is distinct from $\mathbf{Z}[\sqrt{-3}]$).$^8$

Before Dedekind and Kronecker, the only rings studied are still of the type $\mathbf{Z}[\theta]$ or sometimes certain particular rings of the type $\mathbf{Z}[\theta, \theta']$.$^9$

As far as Kronecker is concerned, it is possible that the idea of considering the ring of all integers of an algebraic extension was first suggested to him by the study of fields of algebraic functions, where this ring appears in a natural way as the set of functions that are “finite at a finite distance”: he insists in any case in his memoir of 1881 on the discriminant (written and announced to the Berlin Academy already in 1862) on this characterisation of the “integers” in these fields ([186 a], v. II, pp. 193-236). Dedekind does not give any indication as to the origin of his own ideas on this point, but from his first publications on number fields in 1871, the ring of all integers of such a field plays a fundamental role in his theory; it is also Dedekind who clarifies the relation between such a ring and its subrings having the same field of fractions, by the introduction of the notion of conductor ([79], v. I, pp. 105-157).

But that was not the only difficulty. To generalise the ideas of Kummer, it was first necessary to avoid going through the decomposition field which naturally could not have any analogue in the case of a non abelian field. This detour seems anyway at first sight very surprising and artificial, for if one starts from the irreducible polynomial $\Phi_p(X)$ of $\mathbf{Z}[X]$, one asks oneself why Kummer does not follow to the bitter end the logical consequences of his ideas, and what it is that stops him using the theory of “Galois imaginaries”, well known at that time. The obstacle appears more clearly in the light of an unhappy attempt to generalise made already in 1865 by Selling, a pupil of Dedekind: given an irreducible polynomial $P \in \mathbf{Z}[X]$, Selling decomposes the corresponding polynomial $\overline{P}(X)$ into irreducible factors in $\mathbf{F}_q[X]$; the roots of this polynomial belong therefore to a finite extension $\mathbf{F}_r$ of $\mathbf{F}_q$; but Selling, in order to define in the same way as Kummer the exponent of a “prime ideal factor” of $q$ in an integer of the field of roots of $P(X)$, does not hesitate to speak in the field $\mathbf{F}_r$ of congruences modulo a power of $q$ ([282], p. 26); and a little later, when he tries to tackle the question of ramification, he “adjoins” to $\mathbf{F}_r$ “imaginary roots” of an equation of the form $x^h = q$ ([282], p. 34). It is obvious that these bold steps (which would be justified by replacing the finite field $\mathbf{F}_q$ by the $q$-adic field) could not at that time end up with anything

$^8$ Although Kronecker must have been led to study the arithmetic of the rings $\mathbf{Z}[\sqrt{-D}]$ ($D > 0$) by his work on complex multiplication, he did not publish anything on this subject, and the characterisation of the integers of an arbitrary quadratic field $\mathbf{Q}(\sqrt{D})$ is given explicitly for the first time by Dedekind in 1871 ([79], v. I, pp. 105-106).

$^9$ We saw above the example of the ring $\mathbf{Z}[\zeta, \mu]$ introduced by Kummer [188 e]. Beforehand Eisenstein had been led to consider a subring generated by two elements of the ring of integers in a field of 21st roots of unity [102 b].

except nonsense. Fortunately Dedekind had just in 1857 ([79], v. I, pp. 40-66), under the name of "theory of superior congruences", taken up under another form the theory of finite fields:$^{10}$ he interprets the elements of these latter as "remainders" of polynomials in $\mathbf{Z}[X]$ following a "double module" consisting of linear combinations, with coefficients in $\mathbf{Z}[X]$, of a prime number $p$ and a unitary irreducible polynomial $P \in \mathbf{Z}[X]$ (which is no doubt for him, as for Kronecker, at the beginnings of the general idea of module at which they will independently end up a little later). By his own testimony ([79], v. I, p. 218) it appears that Dedekind had begun to attack the problem of "ideal factors" of $p$ in a field $\mathbf{Q}(\xi)$, where $P \in \mathbf{Z}[X]$ is the minimal polynomial of $\xi$, in the following way (at least in the "unramified" case), that is to say when in $\mathbf{F}_p[X]$ the polynomial $\overline{P}$ corresponding to $P$ does not have multiple roots): we write, in $\mathbf{Z}[X]$,

where the $\overline{P}_i$ are irreducible and distinct in $\mathbf{F}_p[X]$; it can be assumed that $G$ is not divisible (in $\mathbf{Z}[X]$) by any of the $P_i$ and for all $i$, we set $W_i = \prod_{j \neq i} P_j$; then, if $f \in \mathbf{Z}[X]$, we will say that $f(\xi)$ contains the "ideal factor" $p_i$ of $p$ corresponding to $P_i$ $k$ times if we have

and

The relationship with the method followed by Kummer for the "Kummer fields" is manifest here, and one can equally, in this way, rejoin easily the initial definition of Kummer for the cyclotomic fields (see for example the work of Zolotareff [343] who, at first independently of Dedekind, develops these ideas a little later).

However neither Dedekind, nor Kronecker who seems also to have made similar attempts, were to continue further along this path, both stopped by the difficulties presented by ramification ([79], v. I, p. 218 and [186 a], v. II, p. 325).$^{11}$ If the ring of integers $A$ of a number field $K$ which is considered as admitting a basis (over $\mathbf{Z}$) consisting of the powers of the same integer $\theta$, it is not difficult to generalise the preceding method for the ramified prime numbers in $\mathbf{Z}[\theta]$ (as Zolotareff indicates (loc. cit.)). But there are fields $K$ where no basis of this type exists in the ring $A$; and Dedekind even ends up by discovering that there are cases where certain prime numbers $p$ (the "extraordinary factors of the discriminant" of the field $K$) are such that, whatever

$^{10}$ It is known that certain results of that theory, published first by Galois, had been obtained (in the language of congruences) by Gauss in about 1800; after the death of Gauss, Dedekind had set himself the task of publishing part of his work and had in particular found again in the papers left by Gauss the memoir on finite fields ([124 a] v. II, pp. 212-240). $^{11}$ Zolotareff gets round the difficulty by a refinement of his method, which seems no longer to have other than an anecdotal interest [343].

the $\theta \in A$, application of the preceding method to the minimal polynomial of $\theta$ over $\mathbf{Q}$ would lead to attributing to $p$ multiple ideal factors whereas in fact $p$ is not ramified in $A$;$^{12}$ he admits to having been stopped for a long time by this unforeseen difficulty, before succeeding in surmounting it by creating in its entirety the theory of modules and ideals, expounded in a masterly way (and already totally modern, in contrast with the discursive style of his contemporaries) in what is no doubt his masterpiece, the famous "XIth supplement" to the book of Dirichlet on number Theory ([79], v. III, pp. 1-222). This work will know three successive versions, but from the first (published as the "Xth supplement" to the second edition of the book of Dirichlet in 1871) the essentials of the method are already in place, and almost in a single step the theory of algebraic numbers passes from sketches and previous fumblings to a discipline in full maturity and already in possession of its essential tools: from the beginning, the ring of all integers of a number field is placed at the centre of the theory; Dedekind proves the existence of a basis of this ring over $\mathbf{Z}$, and deduces from that the definition of the discriminant of a field, as the square of the determinant consisting of the elements of a basis of the ring of integers and their conjugates; he gives however in the XIth supplement a characterisation of ramified prime numbers (as prime factors of the discriminant) only for quadratic fields ([79] v. III, p. 202), whereas he was in possession of the general theorem since 1871.$^{13}$ The central result of the work is the theorem on the existence and uniqueness of the decomposition of ideals into prime factors, for which Dedekind starts by developing an elementary theory of "modules"; in fact, in the XIth supplement, he reserves this name for the $\mathbf{Z}$-submodules of a number field, but the conception of it that he builds and the results that he proves are already set out in a way that is immediately applicable to the most general modules;$^{14}$ we must note among other things, from 1871, the introduction of the notion of "Noether quotient" which plays an important role (as well besides as the "ascending chain condition") in the first proof of unique factorisation. In the following two editions, Dedekind was to give as well two other proofs of this theorem, which he considered justifiably to be the keystone of this theory. It must be noted here that it is in the third proof that fractional ideals occur (already introduced by Kummer from 1859 for cyclotomic fields) and the fact that they form a group; we will return later to the second proof (p. 107).

$^{12}$ Kronecker says that he had encountered the same phenomenon in a subfield of the field of 13th roots of unity, which he does not go into further however ([186 a], v. II, p. 384). The example of the extraordinary factor of the discriminant given by Dedekind is treated in detail in [150 d], p. 333; a little later, Hasse gives an example of a field $K$ where there is no extraordinary factor of the discriminant, but where there exists no $\theta \in A$ such that $A = \mathbf{Z}[\theta]$ ([150 d], p. 335). $^{13}$ He only published the proof of this theorem in his memoir of 1882 on the different ([79], v. I, pp. 351-396). $^{14}$ In his memoir of 1882 on algebraic curves (in common with H. Weber) [80], he uses in the same way the theory of modules over the ring $\mathbf{C}[X]$.

All these results (up to the language used) were without doubt already known to Kronecker in about 1860 as particular cases of his more general concepts of which we will speak later (whereas Dedekind admits not having overcome the last difficulties of the theory until 1869-1870 ([79], v. I, p. 351));$^{15}$ as far as number fields are concerned, we must underline in particular that, from this time, Kronecker knew that the whole theory is applicable without any essential change when one departs from a "base field" $k$ which is itself a number field (other than $\mathbf{Q}$), a point of view to which the theory of complex multiplication naturally led; he had also recognised, for certain fields $k$, the existence of algebraic extensions $K \neq k$ unramified over $k$ ([186 a], v. II, p. 269) which can not happen for $k = \mathbf{Q}$ (as follows from the output of Hermite and Minkowski for the discriminant). Dedekind was never to develop this last point of view (although he indicates its possibility in his memoir of 1882 on the different), and the first systematic exposition of the theory of "relative fields" is due to Hilbert ([163 a], v. I, pp. 63-363).

Finally, in 1882 ([79], v. I, pp. 359-396), Dedekind completes the theory by introducing the different, which gives him a new definition of the discriminant and allows him to specify the exponents of the prime ideal factors in the decomposition of the latter. It is also at about this time that he becomes interested in the particular points brought out by Galois extensions, introducing the notions of a decomposition group and an inertia group (in a memoir ([79], v. II, pp. 43-48) that was only published in 1894), and even (in papers unpublished during his lifetime ([79], v. II, pp. 410-411)) a sketch of ramification groups, that Hilbert (independently of Dedekind) would develop a little later ([163 a], v. I, pp. 13-23 and 63-363).

So, in about 1895, Algebraic Number theory had finished the first stage of its development; the tools forged during this period of formation are going to allow it to tackle almost straightaway the next stage, the general theory of class fields (or, what amounts to the same thing, the theory of abelian extensions of number fields) which is carried on to our day and which we will not need to describe here. From the point of view of Commutative Algebra, it can be said that at the same time the theory of Dedekind rings is practically completed, apart from their axiomatic characterisation, as well as the structure of modules of finite type over these rings (which, in the case of number fields, will only be clarified in substance by Steinitz in 1912 [294 b]).$^{16}$

The subsequent progress of Commutative Algebra is going to arise above all from quite different problems, issuing from Algebraic Geometry (which,

$^{15}$ Kronecker had however not succeeded in obtaining by his methods the complete characterisation of ramified ideals in the case of number fields. On the other hand, he has this characterisation for the field of algebraic functions of a variable, and proves as well that in this case there is no "extraordinary factor" of the discriminant ([186 a], v. II, pp. 193-236).

$^{16}$ A beginning of the study of modules over a field of algebraic integers had already been started by Dedekind ([79], v. II, pp. 59-85).

besides will influence in a direct way Number Theory, even before the "abstract" developments of the contemporary era).

We are not going to do the detailed history of Algebraic Geometry here, which, until the death of Riemann, hardly touches on our subject. It is sufficient to remember that it had above all as its goal the study of algebraic curves in the complex projective plane, tackled most often by the methods of projective geometry (with or without the use of co-ordinates). In parallel there was developing, with Abel, Jacobi, Weierstrass and Riemann, the theory of "algebraic functions" of one complex variable and their integrals; the link between this theory and the geometry of plane algebraic curves was evidently in their minds, and it was known how on that occasion to "apply Analysis to Geometry"; but the methods used for the study of algebraic functions were mainly of a "transcendental" nature, even before Riemann;$^{17}$ this characteristic is more accentuated in the work of this latter, with the introduction of "Riemann surfaces" and of arbitrary analytic functions defined on such a surface. Almost immediately after the death of Riemann, Roch and especially Clebsch, recognised the possibility of drawing out from the deep results obtained by the transcendental methods of Riemann numerous and striking applications to the projective geometry of curves, which was naturally going to encourage contemporary geometers to give these results purely "geometric" proofs; this programme, incompletely followed by Clebsch and Gordan, was accomplished by Brill and M. Noether a few years later [37], by means of the study of systems of variable points on a given curve and of auxiliary curves (the "adjoins") going through such systems of points. But even for the contemporaries, the transcendental methods of Riemann (and notably his use of topological notions and the "Dirichlet principle") seem to rest on uncertain foundations; and even though Brill and Noether were more careful than the majority of contemporary "synthetic" geometers (see later p. 106), their geometrico-analytic arguments are not safe from all blame. It is essentially in order to give to the theory of plane algebraic curves a solid basis that Dedekind and Weber publish in 1882 their great memoir on this subject [80]: "The researches published herewith", they say, "have as their goal to state the foundations of the theory of algebraic functions of one variable, one of the main creations of Riemann, in a way that is at the same time simple, rigorous and entirely general. In later researches on this subject,

$^{17}$ It must be noted however that Weierstrass; in his research on abelian functions (which go back to 1857 but were only brought out in his lectures in about 1865 and published only in his Complete Works ([329 a], v. IV)), gives as opposed to Riemann, a purely algebraic definition of the genus of a curve as the least integer $p$ such that there are rational functions on the curve having poles at $p+1$ arbitrary given points. It is interesting to point out that, seeking to obtain elements which take the place for him of functions having only a single pole on the curve, Weierstrass, before finally using for this purpose transcendental functions, had, as witnessed by Kronecker ([186 a], v. II, p. 197), encouraged this latter to extend to algebraic functions of one variable the results that he had just obtained at that time concerning number fields (the "prime ideal factors" playing effectively the role desired by Weierstrass).

restrictive hypotheses on the singularities of the functions being considered are often made, and the supposed exceptional cases are, either mentioned in passing as limit cases, or entirely neglected. In the same way, certain fundamental theorems on continuity or analyticity are assumed, for which the "evidence" leans on geometric intuitions of a varied nature" ([80], p. 181).18 The essential idea of their work is to copy the theory of algebraic functions of one variable onto the theory of algebraic numbers such as it had just been developed by Dedekind; to do this, they must first put themselves in an "affine" point of view (over against their contemporaries who invariably considered algebraic curves as embedded in complex projective space): they start therefore from a finite algebraic extension $K$ of the field $\mathbf{C}(X)$ of rational fractions, and from the ring $A$ of "integral algebraic functions" in $K$, i. e. from the elements of this field which are integers over the ring $\mathbf{C}[X]$ of polynomials; their fundamental result, that they obtain without using any topological consideration,19 is that $A$ is a Dedekind ring, to which are applied mutatis mutandis (and even, as Dedekind and Weber remark without having seen clearly the reason yet ([79], v. I, p. 268), in a simpler way) all the results of the "XIth supplement". That done, they prove that their theorems are in fact birationally invariant (in other words, only depending on the field $K$) and in particular do not depend on the choice of the "line at infinity" made at the beginning. What is no doubt even more interesting for us, is that, wanting to define the points of the "Riemann surface" corresponding to $K$ (and in particular the "points at infinity", which could not correspond to ideals of $A$), they are led to introduce the notion of a "place" of the field $K$: they find themselves faced with the situation that will be found again by Gelfand in 1940 while setting up the theory of normed algebras, namely a set $K$ of elements which are not given in advance as functions, but that nonetheless one wants to be able to consider as such; and, in order to obtain the set of definition of these hypothetical functions, they have for the first time the idea (that Gelfand will take up again, and which has become dull through having been used at every turn in modern mathematics) of associating with a point $x$ of a set $E$ and with a set $\mathscr{F}$ of maps of $E$ into a set $G$ the map

18 It is known that, in spite of the efforts of Dedekind, Weber and Kronecker, the relaxation in the concept of what constitutes a correct proof, already noticeable in the German school of Geometry in the years 1870-1880, was only to become worse and worse in the work of the French geometers and especially the Italians of the following two generations, who, following the German geometers, and in developing their methods, attack the theory of algebraic surfaces: a "scandal" denounced several times (especially from 1920) by the algebraists, but which was not without being justified to a certain extent by the brilliant successes obtained by these "non-rigorous" methods, contrasting with the fact that, until about 1940, the orthodox successors of Dedekind showed themselves incapable of formulating with enough flexibility and power the algebraic notions that would have allowed them to give correct proofs for these results. 19 They underline that, thanks to this fact, all these results remain valid when replacing the field $\mathbf{C}$ with the field of all algebraic numbers ([79], v. I, p. 240).

$f \mapsto f(x)$ of $\mathscr{F}$ into $G$, in other words to consider, in the expression $f(x)$, $f$ as variable and $x$ as fixed, contrary to the classical tradition. Finally, they have no trouble, starting from the notion of place, in defining the "positive divisors" ("Polygone" in their terminology) that refine the notion of ideal of $A$ and correspond to the "systems of points" of Brill and Noether; but, although they write the principal divisors and the divisors of differentials as "quotients" of positive divisors, they do not give the general definition of divisors, and it is only in 1902 that Hensel and Landsberg will introduce, in analogy with fractional ideals, this notion that will always embarrass the holders of purely "geometric" methods (obliged in spite of themselves to define them under the name of "virtual systems", but hindered from being able to give them a "concrete" interpretation).

The same year 1882 also sees the appearance of the great memoir of Kronecker awaited for more than 20 years ([186 a], v. II, pp. 237-387). Much more ambitious than the work of Dedekind-Weber, it is unhappily also much more vague and obscure. His central theme is (in modern language) the study of the ideals of a finite integral algebra over one of the rings of polynomials $\mathbf{C}[X_1, \ldots, X_n]$ or $\mathbf{Z}[X_1, \ldots, X_n]$; Kronecker limits himself $a$ priori to those of these ideals that are of finite type (the fact that they are all of this kind was only to be proved (for the ideals of $\mathbf{C}[X_1, \ldots, X_n]$ several years later by Hilbert during his work on invariants ([163 a], v. II, pp. 199-257)). As far as $\mathbf{C}[X_1, \ldots, X_n]$ or $\mathbf{Z}[X_1, \ldots, X_n]$ are concerned one was naturally led to associate to each ideal of one of these rings the "algebraic variety" consisting of the common zeros of all the elements of the ideal; and the geometrical studies in 2 or 3 dimensions done during the course of the XIXth century would intuitively lead to the idea that every variety is the union of "irreducible" varieties, finite in number, whose "dimensions" are not necessarily the same. It appears that the proof of this fact was the goal that Kronecker was setting himself, although he does not state it explicitly anywhere, and that nowhere in his memoir can any definition of "irreducible variety" nor of "dimension" be found. In fact he limits himself to indicating summarily how a general method of elimination$^{20}$ gives, starting with a system of generators of the ideal under consideration, a finite number of algebraic varieties, for each of which, in a suitable system of co-ordinates, a certain number of the

$^{20}$ By a linear change of co-ordinates, one can assume that the generators $F_i$ ($1 \leq r \leq r$) of the ideal are polynomials where the term of highest degree in $X_1$ is of the form $c_i X_1^{m_i}$, where $c_i$ is a constant $\neq 0$. It can also be assumed that the $F_i$ do not have a common factor. One considers then for $2r$ indeterminates $u_i, v_i$ ($1 \leq i \leq r$) the polynomials $\sum_{i=1}^r u_i F_i$ and $\sum_{i=1}^r v_i F_I$ as polynomials in $X_i$; one calculates their Sylvester resultant, which is a polynomial in the $u_i$ and $v_i$, with coefficients in $\mathbf{C}[X_1, \ldots, X_n]$ (respectively $\mathbf{Z}[X_1, \ldots, X_n]$); in cancelling these coefficients, a system of equations is obtained whose solutions $(x_2, \ldots, x_n)$ are exactly the projections of the solutions $(x_1, \ldots, x_n)$ of the system of equations $F_i(x_1, x_2, \ldots, x_n) = 0$ ($1 \leq i \leq r$). One can then continue with the application of the method by induction on $n$.

co-ordinates are arbitrary and the others are "algebraic functions". But, if it is really the decomposition into irreducible varieties that Kronecker is aiming at, it has to be recognised that he only succeeds in the elementary case of a principal ideal, where he proves substantially, by extending a classical lemma of Gauss on $\mathbf{Z}[X]$ ([124 a], v. I, p. 34), that the rings $\mathbf{C}[X_1, \ldots, X_n]$ and $\mathbf{Z}[X_1, \ldots, X_n]$ are unique factorisation domains; and, in the general case, it can even be asked if Kronecker was at that time in possession of the notion of prime ideal (what he calls "Primmodulsystem" is an ideal indecomposable as the product of two others ([186 a], v. II, p. 336); that is all the more astonishing as the definition given since 1871 by Dedekind was perfectly general). A little later, Kronecker was however to correct this inadvertence.

It must be said however that Kronecker's method of elimination, suitably applied, leads exactly to the decomposition of an algebraic variety into its irreducible components: that is what is clearly established by E. Lasker at the beginning of his great memoir of 1905 on ideals of polynomials [194]; he defines correctly the notion of irreducible variety (in $\mathbf{C}^n$) as an algebraic variety $V$ such that the product of two polynomials can not be zero in the whole variety $V$ unless one of them is zero on it, and there is also a definition of dimension that is independent of the axes chosen. In the interesting historical considerations that he inserts in this work, Lasker indicates on the other hand that he links up, not only with the purely algebraic tendency of Kronecker and Dedekind, but also with the problems raised by the geometric methods of the school of Clebsch and M. Noether, and notably with the famous theorem proved by the latter in 1873 [239]. It consists essentially, as we would say today, of the determination of the ideal $a$ of those polynomials of $\mathbf{C}[X_1, \ldots, X_n]$ which are zero on the points of a given set $M$ in $\mathbf{C}^n$; most often $M$ was the "algebraic variety" of the zeros common to a finite number of polynomials $f_i$, and for a long time it appears that it was assumed (of course without justification) that, at least for $n = 2$ or $n = 3$, the ideal $a$ was quite simply generated by the $f_i$; M. Noether had shown that already for $n = 2$ and for two polynomials $f_1, f_2$, this is in general inexact, and he had given sufficient conditions for $a$ to be generated by $f_1$ and $f_2$. Ten years later, Netto proves that, without any hypothesis on $f_1$ and $f_2$, a power of $a$ is in any case contained in the ideal generated by $f_1$ and $f_2$ [231], a theorem that Hilbert generalises in 1893 in his famous "zeros theorem" ([163 a], v. II, pp. 287-344). It is without doubt that, inspired by this result Lasker, in his memoir, intro-

21 It is the number of arbitrary co-ordinates that he calls the dimension ("Stufe"). 22 See the remarks of M. Noether at the beginning of his memoir [239]. It is interesting to note on this point that, according to Lasker, Cayley, in about 1860, had conjectured that for every left algebraic curve in $\mathbf{C}^3$, there was a finite number of polynomials generating the ideal of polynomials of $\mathbf{C}[X, Y, Z]$ which are zero on the curve (in other words, a particular case of the finiteness theorem of Hilbert ([163 a], v. II, pp. 199-257)).

duces the general notion of primary ideal$^{23}$ in the rings $\mathbf{C}[X_1, \ldots, X_n]$ and $\mathbf{Z}[X_1, \ldots, X_n]$ (after having given in these rings the definition of prime ideals, by transcribing the definition of Dedekind), and proves$^{24}$ the existence of a primary decomposition for every ideal in these rings.$^{25}$ He does not seem to have been concerned with questions of uniqueness of this decomposition; it is Macaulay who, a little later [211] introduces the distinction between "immersed" and "non-immersed" primary ideals and shows that the latter are determined in a unique way, but not the earlier. It must finally be noted that Lasker also extends his results to the ring of entire series convergent in the neighbourhood of a point, relying on the "preparation theorem" of Weierstrass. This part of his memoir is doubtless the first place where this ring had been considered from a purely algebraic point of view, and the methods that Lasker develops for this purpose were to influence Krull strongly when in 1938 he created the general theory of local rings (cf. [187 h], pp. 204 and passim).

The movement of ideas that will end up with modern Commutative Algebra starts to take shape around 1910. If the general notion of a field is acquired already at the beginning of the XXth century, on the other hand the first work where the general notion of a ring is defined is without doubt that of Fraenkel in 1914 [113 a]. At that time, there were already as examples of rings, not only the integral rings of Number Theory and of Algebraic

$^{23}$ Examples of primary ideals which are not powers of prime ideals had been encountered by Dedekind in "orders", i.e. the rings of algebraic numbers having a given number field as field of fractions ([79], v. III, p. 306). Kronecker also gives as an example of an ideal "indecomposable" into a product of two others which are non trivial, the ideal of $\mathbf{Z}[X]$ generated by $p^2$ and $X^2 + p$, where $p$ is a prime number (an ideal that is primary for the prime ideal generated by $X$ and $p$ ([186 a], v. II, p. 341)). $^{24}$Lasker proceeds by induction on the maximal dimension $h$ of the irreducible components of the variety $V$ of zeros of the ideal $a$ under consideration. In modern terms, he considers first the prime ideals $p_i$ ( $1 \leq i \leq r$) containing $a$ which correspond to the irreducible components of maximal dimension $h$ of $V$. To each $p_i$, he associates the saturated $q_i$ of $a$ relative to $p_i$; he considers then the Noether quotient $b_i = a : q_i$ of $q_i$ in $a$, takes in $\sum_i b_i$ an element $c$ not belonging to any of the $p_i$ and shows on the one hand that $a$ is the intersection of the $q_i$ and of $a + (c) = a'$, and on the other hand that the variety $V'$ of zeros of $a'$ only has irreducible components of dimension $\leq h - 1$, which enables him to conclude by induction. $^{25}$It is interesting to remark that the second proof of Dedekind for the unique decomposition theorem proceeds by establishing first the existence of a unique reduced primary decomposition; and in a part not published in the XIth supplement, Dedekind observes explicitly that this part of the proof is valid not only for the ring $A$ of all integers in the number field $K$, but also for all the "orders" of $K$ ([79], v. III, p. 303). It is only later, after having proved explicitly that $A$ is "completely integrally closed" (up to terminology) that he shows by using this fact, that the primary ideals of the preceding decomposition are in fact powers of prime ideals ([79], v. III, p. 307).

Geometry, but also rings of series (formal or convergent), and finally algebras (commutative or not) over a base field. However, as much for the theory of rings as for that of fields, the catalytic role seems to have been played by the theory of $p$-adic numbers of Hensel, that Fraenkel as well as Steinitz [294 a] mention quite particularly as the point of departure for their research.

The first publication of Hensel on this subject goes back to 1897; in it he starts from the analogy brought to light by Dedekind and Weber between the points of a Riemann surface of a field of algebraic functions $K$ and the prime ideals of a number field $k$; he proposes to transport into Number Theory the "Puiseux expansions" (classical since the middle of the XIXth century) which, in the neighbourhood of an arbitrary point of a Riemann surface of $K$, allow us to express every element $x \in K$ in the form of a convergent series of powers of the "uniformiser" at the point under consideration (a series having only a finite number of terms with negative exponents). Hensel shows in the same vein that if $\mathfrak{p}$ is a prime ideal of $k$ above a prime number $p$, one can associate to each $x \in k$ a "$p$-adic series" of the form $\sum_i \alpha_i p^i$ (or $\sum_i \alpha_i p^{i/e}$ when $\mathfrak{p}$ is ramified above $p$) the $\alpha_i$ being taken from a system of given representatives of the quotient field of the ideal $\mathfrak{p}$; but his great originality is to have the idea of considering such "expansions" even when they do not correspond to any element of $k$, by analogy with the expansions in entire series of transcendental functions on a Riemann surface [157 a].

During the whole of his subsequent career, Hensel will devote himself to polishing and perfecting little by little his new calculus; and if his approach can seem to us hesitant or heavy, it must not be forgotten that at least in the beginning he still does not have available any of the topological or algebraic tools of present day mathematics which would have facilitated his task. In his first publications he besides hardly ever speaks of topological notions, and to sum up, for him the ring of $p$-adic numbers ($\mathfrak{p}$ a prime ideal of the ring of integers $A$ of a number field $k$), is, in modern terms, the projective limit of the rings $A/p^n$ for $n$ tending to infinity, with a purely algebraic meaning; and to establish the properties of this ring and of its field of fractions, he must at each step use more or less painfully $ad\ hoc$ arguments (for example to prove that the $p$-adic numbers form an integral domain). The idea of introducing topological notions in a $p$-adic field hardly appears with Hensel before 1905 [157 d]; and it is only in 1907, after having finished writing the book where he restates the theory of algebraic numbers following his ideas [157 f], that he reaches the definition and essential properties of absolute $p$-adic values [157 e], starting from which he will be able to develop, by copying the theory of Cauchy, a complete "$p$-adic analysis" that he will know how to apply fruitfully in number Theory (notably with the use of the $p$-adic exponential and logarithm), and whose importance has not ceased to grow since then.

Hensel had well seen, from the beginning, the simplifications that his theory was bringing to classical statements, by allowing the "localisation" of problems and working in a field where not only the properties of divisibility are trivial, but also where, thanks to a fundamental lemma that he got out already in 1902 [157 c], the study of polynomials of which the polynomial "reduced" mod $p$ has no multiple root is reduced to the study of polynomials over a finite field. He had given from 1897 [157 b] striking examples of these simplifications, notably in questions referring to the discriminant (in particular, a short proof of the criterion given by him some years previously for the existence of "extraordinary divisors"). But for a long time it seems as if the $p$-adic numbers inspired a great distrust in contemporary mathematicians; an attitude current without doubt towards ideas that were too "abstract", but that the slightly excessive enthusiasm of their author (so frequent in mathematics amongst the zealots of new theories) was not without justifying in part. Not content in fact with applying his theory fruitfully to algebraic numbers, Hensel, impressed like all his contemporaries by the proofs of transcendence of $e$ and $\pi$, and perhaps deluded by the qualification of "transcendent" applied at the same time to numbers and to functions, had reached the point of thinking that there existed a link between his $p$-adic numbers and real transcendent numbers, and he had believed for a moment that he could obtain thus a simple proof of the transcendence of $e$ and even of $e^e$ ([157 d], p. 556).26

Soon after 1910, the situation changes, with the rising of the following generation, influenced by the ideas of Fréchet and of F. Riesz on topology, by those of Steinitz on algebra, and conquered by "abstraction" from the beginning; it will know how to make assimilable and put in its place the work of Hensel. From 1913, Kürschak [190] defines in a general way the notion of valuation, recognises the importance of the archimedean valuations (of which the $p$-adic valuation gave an example), proves (by copying the proof in the case of real numbers) the existence of the completion of a field with respect to a valuation, and above all proves in a general way the possibility of the extension of a valuation to an algebraic extension of a given field. But he had not seen that the non-archimedean character of a valuation was already hidden in the prime field; that point was established by Ostrowski, to whom is also due the determination of all the valuations on the field $\mathbf{Q}$, and the fundamental theorem characterising fields furnished with a non archimedean valuation as subfields of $\mathbf{C}$ [242]. In the years that go from 1920 to 1935, the theory will complete itself with a more detailed study of valuations which are not necessarily discrete, including among others the examination of the different circumstances which arise when one goes to an algebraic or transcendental extension (Ostrowski, Deuring, F. K. Schmidt); on the other hand, in 1931, Krull introduces and studies the general notion of valuation

26 This research at all costs for a strict parallelism between $p$-adic series and Taylor series also pushes Hensel to set himself strange problems: he proves for example that every $p$-adic integer can be written in the form of a series $\sum_{k=0}^{\infty} a_k p^k$ where the $a_k$ are rational numbers chosen in such a way that the series converges not only in $\mathbf{Q}_p$ but also in $\mathbf{R}$ (no doubt by remembering Taylor series that converge at several points simultaneously?) [157 e and f].

[187 f] which will be much used in the years that follow by Zariski and his school of algebraic Geometry.27 We must also mention here, even though it is outside our remit, the deeper studies on the structure of complete valuation fields and complete local rings, which come from the same period (Hasse-Schmidt, Witt, Teichmüller, I. Cohen).

The work of Fraenkel mentioned above (p. 107) dealt only with a very special kind of ring (artinian rings having only a single prime ideal, which is, as well, assumed to be principal). If one excludes the work of Steinitz on fields [294 a], the first important work on the study of general commutative rings are the two great memoirs of E. Noether on the theory of ideals: that of 1921 [236 a], dedicated to the primary decomposition, which takes up again on the most general level and completes on many points the results of Lasker and Macaulay; and that of 1927 characterising Dedekind rings axiomatically [236 b]. In the same way that Steinitz had shown it for fields, it is seen in these memoirs how a small number of abstract ideas, such as the notion of irreducible ideal, chain conditions and the idea of an integrally closed ring (the last two, as we have seen, already brought out by Dedekind) can on their own lead to general results which seem inextricably linked to results of pure calculation in the cases where they were known beforehand.

With these memoirs of E. Noether, joined to the slightly later work of Artin-van der Waerden on divisor ideals ([317 a], v. II, pp. 105-109) and of Krull linking these ideals to essential valuations [187 f] is thus completed the long study of the decomposition of ideals started a century before,28 at the same time as modern Commutative Algebra is inaugurated.

The countless later researches on Commutative Algebra are grouped most easily following some major directing tendencies:

A) Local and topological rings. Although contained in embryo in all the previous work in Number Theory and Algebraic Geometry, the general idea of localisation frees itself very slowly. The general notion of a ring of fractions is only defined in 1926 by H. Grell, a pupil of E. Noether, and only for integral domains [137]; its extension to more general rings will only be given in 1944 by C. Chevalley for noetherian rings and in 1948 by Uzkov in the general case. Until about 1940, Krull and his school are practically alone in using in general arguments consideration of the local rings $A_p$ of an integral domain;

27 An example of a valuation of height 2 had already been introduced incidentally by H. Jung [175] in 1925. 28 Following the definition of divisor ideals, some fairly numerous researches (Prüfer, Krull, Lorenzen, etc.) were undertaken on ideals that are stable under other operations $a \mapsto a'$ satisfying axiomatic conditions analogous to the properties of the operation $a \mapsto A : (A : a)$ which gives rise to divisor ideals; the results obtained in this way have not found any applications until now in Algebraic Geometry or in Number Theory.

these rings will only begin to appear explicitly in Algebraic Geometry with the work of Chevalley and Zariski starting in 1940.$^{29}$

The general study of local rings themselves only starts in 1938 with the great memoir of Krull [187 h]. The most important results of this work concern dimension theory and regular rings, of which we should not speak here; but it is there also that appears for the first time the completion of an arbitrary local noetherian ring, as well as a still imperfect form of the graded ring associated with a local ring;$^{30}$ this latter will only be defined in about 1948 by P. Samuel [270] and independently by the research in algebraic Topology of Leray and H. Cartan. Krull, in the previously quoted work, hardly uses topological language; but from 1928 [187 e], he had proved that, in a noetherian ring $A$, the intersection of powers of a single ideal $a$ is the set of $x \in A$ such that $x(1-a) = 0$ for an $a \in a$; it is easily deduced from that that for every ideal $m$ of $A$, the $m$-adic topology on $A$ induces on an ideal $a$ the $m$-adic topology of $a$; in his memoir of 1938, Krull completes this result by proving that in a local noetherian ring, every ideal is closed. These theorems were soon after extended by Chevalley to semi-local noetherian rings, then by Zariski to the rings which bear his name [340 b]; it is also to Chevalley that the introduction of "linear compactness" in topological rings goes back, as well as the determination of the structure of semi-local complete rings [62 c].

B) Passing from local to global. Since Weierstrass, the habit has existed of associating an analytic function of one variable (and in particular an algebraic function) with the set of its "expansions" at all the points of the Riemann surface where it is defined. In the introduction to his book on Number Theory ([157 f], p. V), Hensel associates in the same way with every element of an algebraic number field $k$ the set of elements which correspond to it in the completions of $k$ for all valuations$^{31}$ on $k$. It can be said that it is this point of view that, in modern Commutative Algebra, has replaced the decomposition formula for an ideal as a product of prime ideals (extending in a certain

$^{29}$ In the work of Hensel and his pupils on Number Theory, the local rings $A_p$ are systematically neglected for the sake of their completions, no doubt because of the possibility of applying Hensel's lemma to these latter.

$^{30}$ If $m$ is the maximal ideal of the local noetherian ring $A$ under consideration, $(\alpha_i)_{1 \leq i \leq r}$ a minimal system of generators for $m$, Krull defines for all $x \neq 0$ in $A$ the "initial forms" of $x$ in the following way: if $j$ is the largest integer such that $x \in m^j$, the initial forms of $x$ are all the homogeneous polynomials of degree $j$, $P(X_1, \ldots, X_r)$, with coefficients in the quotient field $k = A/m$, such that $x \equiv P(\alpha_1, \ldots, \alpha_r) \pmod{m^{j+1}}$. To every ideal $a$ of $A$ he puts into correspondence the graded ideal of $k[X_1, \ldots, X_r]$ generated by the initial forms of all the elements of $a$ ("Leitdeal"); these two notions take the place for him of the associated graded ring.

$^{31}$ Hensel takes, as non archimedean valuations on a field $K$ of degree $n$ over $\mathbf{Q}$ the functions $x \mapsto |x^{(i)}|$ (where the $x^{(i)}$ for $1 \leq i \leq n$ are the conjugates of $x$) currently used since Dirichlet; Ostrowski will show a little later that these functions are essentially the only non archimedean valuations on $K$.

sense the initial point of view of Kummer). The remark of Hensel reduces implicitly to embedding $k$ in the product of all its completions; that is what is done explicitly by Chevalley in 1936 with his theory of "idèles" [62 b], which perfects the previous analogous ideas of Prüfer and von Neumann (these latter limit themselves to embedding $k$ in the product of its $p$-adic completions).$^{32}$ Although this is a bit outside our remit, it is important to mention here that, thanks to an appropriate topology on the group of idèles, all the techniques of locally compact groups (including Haar measure) can thus be applied to Number Theory in a very effective way.

In a more general setting, the theorem of Krull [187 f], characterising an integrally closed ring as the intersection of valuation rings (which comes back to embedding the ring under consideration in the product of valuation rings) often eases the study of these rings, although the method is not really manageable except for the essential valuations of Krull rings. Examples (fairly elementary) are besides often found in Krull [187 i] of the method of "passing from the local to the global" consisting of proving a property of an integral domain $A$ by reducing it to verifying it for the "localisations" $A_p$ of $A$ at all its prime ideals;$^{33}$ more recently, Serre has noticed that this method is valid for arbitrary commutative rings $A$, that it is applicable also to $A$-modules and to their homomorphisms and that it is sufficient often to "localise" at the maximal ideals of $A$: a point of view that is tied closely to ideas on "spectra" and on sheaves defined on these spectra (see below, p. 115).

C) Integers and integral closure. We have seen that the notion of algebraic integer, first introduced for number fields, had already been extended by Kronecker and Dedekind to algebraic function fields, although in this case it could appear fairly artificial (not corresponding to a projective notion). The memoir of E. Noether of 1927, followed by the work of Krull starting in 1931, was to show the interest that these notions were to offer for the most general rings.$^{34}$ It is due to Krull in particular that we owe the lifting theorems for

$^{32}$ Because of this remark of Hensel, the habit has arisen (by abuse of language) of calling the non archimedean valuations of $K$ the "places at infinity" of a number field $K$, by analogy with the process by which Dedekind and Weber define the "points at infinity" of the Riemann surface of an affine curve (cf. p. 104). $^{33}$ When one speaks of "passing from the local to the global", reference is often made to much more difficult questions, linked to the theory of class fields, and of which the best known examples are those treated in the memoirs of Hasse [150 a and b] on quadratic forms over an algebraic number field $k$; he shows there among other things that for an equation $f(x_1, \ldots, x_n) = a$ to have a solution in $k^n$ ($f$ a quadratic form, $a \in k$), it is necessary and sufficient that it has a solution in each of the completions of $k$. By the testimony of Hasse, the idea for this type of theorem was suggested by his master Hensel [150 c]. The extension of this "Hasse principle" to groups other than the orthogonal group is one of the objectives of the modern theory of "adélises" of algebraic groups. $^{34}$ Krull and E. Noether limit themselves to integral domains, but the extension of their methods to the general case is not difficult; the most interesting memoir in this context is that where I. Cohen and Seidenberg extend the theorems of Krull, by indicating exactly their limits of validity [66]. It is appropriate to note that prime ideals in entire algebras [187 g], as well as the extension of the theory of decomposition and inertia groups of Dedekind-Hilbert [187 f]. As for E. Noether, to her is owed the general formulation of the normalisation lemma$^{35}$ (from which follows among others the zeros theorem of Hilbert) as well as the first general criterion (transcription of classical arguments of Kronecker and Dedekind) allowing the affirmation that the integral closure of an integral domain is finite over the ring.

Finally, it must be pointed out here that one of the reasons for the modern importance of the notion of integrally closed ring is due to the studies of Zariski on algebraic varieties; he has discovered in effect that the "normal" varieties (that is to say those whose local rings are integrally closed) stand out by means of particularly pleasant properties, notably the fact they have no "singularity of codimension 1"; and it was noticed afterwards that analogous phenomena take place for "analytical spaces". Also "normalisation" (that is to say the operation that, for local rings of a variety, consists in taking their integral closure) has itself become a powerful tool in the arsenal of modern Algebraic Geometry.

D) The study of modules and the influence of Homological Algebra. One of the distinguishing characteristics of the work of E. Noether and W. Krull in Algebra is the tendency to "linearisation", extending the analogous direction imprinted on field theory by Dedekind and Steinitz; in other words, it is as modules that ideals are considered above all else, and one is therefore led to apply to them all the constructions of linear Algebra (quotient, product, and more recently tensor product and forming homomorphism modules) resulting in general in modules that are no longer ideals. It appears thus fairly quickly that in many questions (whether it is a matter of commutative or non commutative rings), it is not in our interest to limit ourselves to the study of the ideals of a ring $A$, but that it is necessary, on the contrary to state the theorems more generally for $A$-modules (eventually subject to certain finiteness conditions).

The intervention of Homological Algebra only reinforced the previous tendency, since this branch of Algebra is essentially concerned with questions of a linear nature. We should not retrace its history here; but it is interesting to point out that several of the fundamental notions of Homological Algebra (such as those of projective module and that of the functor Tor) were born as a result of the close study of the behaviour of modules over a Dedekind ring relative to the tensor product, a study undertaken by H. Cartan in 1948.

Conversely, it could be foreseen that the new classes of modules introduced in a natural way by Homological Algebra as "universal annihilators" of the functors Ext (projective modules and injective modules) and of the functors Tor (flat modules) would throw a new light on Commutative Algebra. It

E. Noether had explicitly mentioned the possibility of such generalisations in her memoir of 1927 ([236 b], p. 30). $^{35}$ A particular case had already been stated by Hilbert in 1893 ([163 a], v. II, p. 290).

turns out that it is especially the projective modules and even more the flat modules that have proved to be useful: the importance of these latter arises above all from the remark, first made by Serre [283 b], that localisation and completion naturally introduce flat modules, "explaining" thus in a much more satisfactory way the properties of these two operations that were already known and making their use much easier. It is appropriate to mention besides that the applications of Homological Algebra are far from being limited to that, and that it plays a deeper and deeper role in Algebraic Geometry.

E) The notion of spectrum. The last by date of the new notions of Commutative Algebra has a complex history. The spectral theorem of Hilbert introduced ordered sets of orthogonal projectors of a Hilbert space, making up a "Boolean algebra" (or better a Boolean lattice),$^{36}$ in bijective correspondence with a Boolean lattice of classes of measurable subsets (for an appropriate measure) of $\mathbf{R}$. It is no doubt his earlier work on operators in Hilbert spaces that, around 1935, led M. H. Stone to study in a general way Boolean lattices, and notably to look for "representations" of them by subsets of a set (or classes of subsets under a certain equivalence relation). He observes that a Boolean lattice becomes a commutative ring (of a very special type besides), when multiplication is defined by $xy = \inf(x, y)$ and addition by $x + y = \sup(\inf(x, y'), \inf(x', y))$. In the particular case where one leaves the Boolean lattice $\mathfrak{P}(X)$ of all subsets of a finite set $X$, it is seen immediately that the elements of $X$ are in a natural bijective correspondence with the maximal ideals of the corresponding "Boolean" ring; and Stone obtains exactly his general theorem about the representation of a Boolean lattice on the set of maximal ideals that contain it [301 a].

On the other hand, it was known that a classical example of a Boolean lattice was the set of subsets which are both open and closed in a topological space. In a second work [301 b], Stone showed that in fact every Boolean lattice is also isomorphic to a Boolean lattice of this kind. For that it was naturally necessary to define a topology on the set of maximal ideals of a "Boolean" ring; which is done very simply by taking as closed sets, for each ideal $a$, the set of maximal ideals containing $a$.

We should not speak here of the influence of these ideas in Functional Analysis, where they played an important role in the birth of the theory of normed algebras developed by I. Gelfand and his school. But in 1945, Jacobson observes [172 c] that the procedure of defining a topology, thought up by Stone, can in fact be applied to every ring $A$ (commutative or not) so long as one takes as set of ideals not the set of maximal ideals, but the set of two sided "primitive" ideals (i.e. the two sided ideals $b$ such that $A/b$ is a primitive ring); for a commutative ring, one obtains of course the maximal ideals. From his side, Zariski in 1944 [340 a], uses an analogous

$^{36}$ A Boolean lattice is an ordered lattice $E$, having a least element $\alpha$ and a greatest element $\omega$, where each of the laws sup and inf is distributive with respect to the other and where, for each $a \in E$, there exists one and only one $a' \in E$ such that $\inf(a, a') = \alpha$ and $\sup(a, a') = \omega$.

method for defining a topology on the set of places of an algebraic function field. However these topologies remained simple curiosities for the majority of algebraists, because of the fact that they are in general not Hausdorff, and that there was a fairly understandable reluctance to work with such awkward objects. This mistrust was only dissipated when A. Weil showed, in 1952, that every algebraic variety can be furnished in a natural way with a topology of the preceding type and that this topology allows the definition, in a perfect analogy with the case of differential or analytic varieties, of the notion of a fibred space [330 e]; soon after Serre had the idea of extending to these varieties thus topologised the theory of coherent sheaves, thanks to which the topology supplies in the case of "abstract" varieties the same services as the usual topology when the base field in C, notably in what concerns the application of the methods of Algebraic Topology [283 a and b].

From then on it was natural to use this geometric language in all Commutative Algebra. It was rapidly observed that the consideration of maximal ideals is normally insufficient for obtaining convenient statements,$^{38}$ and that the adequate notion is that of the set of prime ideals of the ring, topologised in the same way. With the introduction of the notion of spectrum, there is at our disposal now a dictionary allowing every theorem of Commutative Algebra to be expressed in a geometric language very close to that of the Algebraic Geometry of the Weil-Zariski period; which besides has led immediately to a considerable widening of the framework of this last, in such a way that Commutative Algebra is now hardly more from this point of view than the most elementary part [138 a].

$^{37}$ The inconvenience of being limited to the "maximal spectrum" arises from the fact that if $\phi : A \to B$ is a homomorphism of rings and $n$ is a maximal ideal of $B$, $\phi^{-1}(n)$ is not necessarily a maximal ideal of $A$, whereas for every prime ideal $p$ of $B$, $\phi^{-1}(p)$ is a prime ideal of $A$. Therefore it is not possible in general to associate with $\phi$ in a natural way a mapping of the set of maximal ideals of $B$ into the set of maximal ideals of $A$.

$^{38}$ The inconvenience of being limited to the "maximal spectrum" arises from the fact that if $\phi : A \to B$ is a homomorphism of rings and $n$ is a maximal ideal of $B$, $\phi^{-1}(n)$ is not necessarily a maximal ideal of $A$, whereas for every prime ideal $p$ of $B$, $\phi^{-1}(p)$ is a prime ideal of $A$. Therefore it is not possible in general to associate with $\phi$ in a natural way a mapping of the set of maximal ideals of $B$ into the set of maximal ideals of $A$.