Theorem 2

Let $A$ be a principal ideal domain, let $K$ be its field offructions and let $P$ be a system of representatives of irreducible elements of $A$. Given an element $x \in K$, there exist a finite subset $H$ of $P$, elements $a_0 \in A$ and $a, \in A$ not divisible by $p$ in $A$ ($p \in H$), and integers $s(p) > 0$ ($p \in H$) such that

where $H$ and the $s(p)$ are uniquely determined by these conditions.

Moreover, if $R_p$ denotes a subset of $A$ containing precisely one element of each residue class mod $p$ in $A (p \in P)$, then each $x \in K$ can be uniquely expressed in the form

where $a \in A$ and $r_{ph} \in R_p$ for all $h$ and $p$, and all but finitely many of the $r_{ph}$ are zero.

Consider $K$ as an $A$-module; then $A$ is the submodule generated by 1. The quotient module $K/A$ is the quotient of $K$ by the equivalence relation $x' - x \in A$, which is also written, in the notation of VI, p. 6, as $x \equiv x'$ (mod 1); let $f$ denote the natural homomorphism from $K$ onto $M = K/A$.

The quotient module $M$ is a torsion module, for every element of $M$ has the form $f(a/b)$ ($a \in A, b \in A, b \neq 0$), whence $b f(a/b) = f(a) = 0$. Hence Th. 1 of VII, p. 8 applies. Let $M_p$ ($p \in P$) be the submodule of elements of $M$ annihilated by powers of $p$; then $f^{-1}(M_p)$ is the subring $A_p$ of elements of $K$ of the form $a p^{-n}$ where $a \in A$ and $n \geq 0$ is an integer. The module $M$ is the direct sum of the $M_p$, so every $x \in K$ is congruent mod 1 to an element in the sum of the $A_p$; in other words, formula (1) holds, with the $s(p)$ integers $> 0$, and the $a$, finitely many elements of $A$ such that $a$, is not a multiple of $p$.

We now show that these conditions on the $s(p)$ and the $a$, completely determine $H$ and the $s(p)$. Indeed $H$ is then the set of $p \in P$ such that the component of $f(x)$ in $M_p$ is $\neq 0$. On the other hand if $s$ and $s'$ are two integers $> 0$ such that s \geq s' and if $a$ and $a'$ are elements of $A$ not divisible by $p$ such that $ap^{-s} \equiv a'p^{-s'} \pmod{1}$, then we deduce that $a \equiv a'p^{s-s'} \pmod{p^s}$; if $s > s'$ then $a \equiv 0 \pmod{p}$, contradicting the hypotheses. This argument also shows that each $a_i$ is well defined mod $p^{s(p)}$.

To complete the proof, we note first of all that in each residue class mod $p^i$ in $A$ there exists a unique element of the form $\sum_{h=0}^{s-1} r_h p^h$ with $r_h \in R_p$ for $0 \leq h \leq s-1$. In fact, we proceed by induction on $s$ (the property follows from the definition of $R_p$ for $s = 1$) : let $x \in A$; by hypothesis there is a unique element of the form $\sum_{h=0}^{s-2} r_h p^h$ ($r_h \in R_p$) in the residue class of $x$ mod $p^{s-1}$; then $x - \sum_{h=0}^{s-2} r_h p^h$ is a multiple $ap^{s-1}$ of $p^{s-1}$; now there exists a unique element $r_s$, of $R_p$ such that $a \equiv r_s \pmod{p}$; whence $x \equiv \sum_{h=0}^{s-1} r_h p^h \pmod{p^s}$. To obtain formula (2) it is now enough to apply this fact to each $a_p$ in formula (1). The uniqueness is clear in view of the above.

The following are the most important applications of Th. 2 :

I. The ring A is the ring Z of rational integers, and K = Q. Let P be the set of prime numbers > 0, and for each $p \in P$ let $R_p$ be the interval $(0, p-1)$ in $\mathbf{Z}$. Thus we have the canonical decomposition

with $a \in \mathbf{Z}$, $e_{ph} \in \mathbf{Z}$ and $0 \leq e_{ph} \leq p-1$.

II. The ring A is the ring E[X] of polynomials in one indeterminate over a commutative field E, and K = E(X). Let P be the set of monic irreducible polynomials in $E[X]$ (VII, p. 5). For $p \in P$ we can, by virtue of euclidean division of polynomials (IV, p. 10), take $R_p$ to be the set of polynomials of degree strictly less than that of $p$. Thus we have the decomposition (called canonical) of a rational function $r(X) \in E(X)$:

where $a(X)$ is a polynomial and $v_{ph}(X)$ is a polynomial of degree strictly less than that of $p(X)$, for all $p$ and $h$. In particular, if the field E is algebraically closed, then the $p(X)$ have the form $X - a$ with $a \in E$, and the $v_{ph}(X)$ are thus constants.

Hence we can say that the vector space $E(X)$ over the field E has a basis consisting of the monomials $X^n$ ($n \geq 0$ an integer) and the rational functions of the form $X^m / (p(X))^h$, where $p \in P$ and $h$ and $m$ are integers with $h \geq 1$ and $0 \leq rn < \deg(p)$.