§ 1. Complex numbers, quaternions

1. Complex numbers, quaternions

1. Definition of complex numbers

The polynomial $X^2 + 1$ has no root in $\mathbf{R}$ , because $x^2 + 1 \geq 1$ for all $x \in \mathbf{R}$ ; it is therefore irreducible over $\mathbf{R}$ . [This is a particular case of the analogous result which applies to any ordered field.]

Definition 1

The field* $\mathbf{R}[X]/(X^2 + 1)$ is called the field of complex numbers and is denoted by $\mathbf{C}$ . The canonical image of $X$ in $\mathbf{C}$ is denoted by $i$ , so that $\mathbf{C}$ is obtained from the field $\mathbf{R}$ by algebraic adjunction of the root $i$ of the polynomial $X^2 + 1$ . The elements of $\mathbf{C}$ are called complex numbers.

From an algebraic point of view the importance of the field $\mathbf{C}$ is due to the following fundamental theorem:

Theorem 1

(d'Alembert-Gauss). The field $\mathbf{C}$ of complex numbers is algebraically closed.

For the proof it is enough to establish that (i) every element $\geq 0$ in $\mathbf{R}$ has a square root, and (ii) every polynomial of odd degree with coefficients in $\mathbf{R}$ has at least one root in $\mathbf{R}$ . The first of these assertions has already been proved in Chapter IV, § 3, no. 3. As to the second, if $f(X) = a_0 X^n + a_1 X^{n-1} + \cdots + a_n$ is a polynomial of odd degree $n$ ( $a_0 \neq 0$ ) with real coefficients, we may write $f(x) = a_0 x^n g(x)$ for $x \neq 0$ , where

g(x) = 1 + \frac{a_1}{a_0 x} + \cdots + \frac{a_n}{a_0 x^n}

tends to $+1$ as $x$ tends to $+\infty$ or $-\infty$ . Hence there is a number $a > 0$ such that $f(a)$ has the sign of $a_0$ and $f(-a)$ the sign of $-a_0$ ;

and so by Bolzano’s theorem (Chapter IV, § 6, no. 1, Theorem 2), $f$ has at least one root in $[—a, a]$ .

Remark 1

Theorem 1 can be proved without invoking the theory of ordered fields by using properties of the topology of the field $\mathbf{C}$ , which will be defined below (no. 2); see § 2, Exercise 2 and also the part of this series devoted to algebraic topology, where the theorem of d’Alembert-Gauss will appear as a consequence of results on the degree of a mapping.

Remark 2

Since $\mathbf{C}$ is of degree 2 over $\mathbf{R}$ , it follows that $\mathbf{C}$ is, up to isomorphism, the only algebraic extension of $\mathbf{R}$ other than $\mathbf{R}$ itself, and that there is no field contained in $\mathbf{C}$ which contains $\mathbf{R}$ , other than $\mathbf{R}$ and $\mathbf{C}$ .

We know that $\mathbf{R}$ may be identified with a subfield of $\mathbf{C}$ , and that every $z \in \mathbf{C}$ can be written uniquely in the form $x + iy$ , where $x$ and $y$ are real; $x$ is called the real part of $z$ and is denoted by $\Re(z)$ ; $y$ the imaginary part of $z$ , denoted by $\Im(z)$ . Complex numbers of the form $iy$ ( $y$ real) are called pure imaginary. The relation $x + iy = 0$ ( $x, y$ real) is equivalent to $x = 0$ and $y = 0$ .

Since $i^2 = -1$ , the elements of $\mathbf{C}$ (when given by their real and imaginary parts) satisfy the following rules of calculation:

(x + iy) + (x' + iy') = (x + x') + i(y + y')

(x + iy)(x' + iy') = (xx' - yy') + i(xy' + x'y').

In particular, $(x + iy)(x - iy) = x^2 + y^2 \in \mathbf{R}$ , so that, if $x + iy \neq 0$ ,

\frac{1}{x + iy} = \frac{x}{x^2 + y^2} - i \frac{y}{x^2 + y^2}.

The second root of the polynomial $X^2 + 1$ in $\mathbf{C}$ is $-i$ ; consequently the only automorphism of $\mathbf{C}$ , other than the identity mapping, which leaves all real numbers invariant, is that which maps $z = x + iy$ to $x - iy$ ; the latter is denoted by $\overline{z}$ and (in agreement with the general definitions) is called the complex number conjugate to $z$ . We have

\Re(z) = \frac{1}{2}(z + \overline{z}) \quad \text{and} \quad \Im(z) = \frac{1}{2i}(z - \overline{z}).

By reason of this automorphism, if $f(z)$ is a polynomial with real coefficients, we have $f(\overline{z}) = \overline{f(z)}$ for all $z \in \mathbf{C}$ .

The real number $z \overline{z} = x^2 + y^2$ is called the algebraic norm of $z$ , or simply the norm whenever there is no risk of confusion; it is a real number $\geq 0$ , which vanishes only if $z = 0$ . The real number $\sqrt{z \overline{z}} = \sqrt{x^2 + y^2} \geq 0$ reduces to the absolute value of $z$ when $z$ is real, and we still call it the absolute value of $z$ , and denote it by $|z|$ , when $z$ is any complex number. The relation $|z| = 0$ is equivalent to $z = 0$ . If $z$ and $z'$ are two complex numbers, the conjugate of $zz'$ is $\overline{zz'}$ , hence $|zz'|^2 = zz'\overline{zz'} = |z|^2|z'|^2$ and therefore $|zz'| = |z|\cdot|z'|$ : the absolute value of a product is the product of the absolute values of the factors. In particular, if $z \neq 0$ and $z' = 1/z$ , we have $|1/z| = |1/z|$ .

Finally, for all complex numbers $z, z'$ , we have the triangle inequality

|z + z'| \leq |z| + |z'|.

2. The topology of $\mathbf{C}$

The mapping $(x, y) \to x + iy$ of the real plane $\mathbf{R}^2$ onto $\mathbf{C}$ is bijective; by means of this bijection we can transport to $\mathbf{C}$ the topology of $\mathbf{R}^2$ (cf. Chapter VI, § 1, no. 5). The topology thus defined on $\mathbf{C}$ is compatible with the field structure of $\mathbf{C}$ (Chapter III, § 6, no. 7), because it is compatible with the ring structure of $\mathbf{C}$ (Chapter VI, § 1, no. 5) and, by (3), $1/z$ is continuous on the complement $\mathbf{C}^*$ of 0 in $\mathbf{C}$ .

If we endow the set $\mathbf{C}$ with this topology and with the field structure defined earlier (no. 1, Definition 1), we have defined on $\mathbf{C}$ the structure of a topological field (Chapter III, § 6, no. 7); whenever we speak of the topology of $\mathbf{C}$ it will always be the above topology that is meant.

In the future we shall generally identify the sets $\mathbf{C}$ and $\mathbf{R}^2$ considered as topological spaces; the subfield $\mathbf{R}$ of $\mathbf{C}$ is then identified with the abscissa of $\mathbf{R}^2$ , which for this reason is called the real axis; likewise the ordinate is called the imaginary axis (note that this is not a subfield of $\mathbf{C}$ ). The ray with origin 0 and direction ratios $(1, 0)$ (identified with $\mathbf{R}_+$ ) is called the positive real semi-axis; the opposite ray, with the same origin and direction ratios $(-1, 0)$ , is called the negative real semi-axis.

For the purposes of graphical illustration we use the representation of $\mathbf{R}^2$ (well known in elementary analytic geometry) by the points of a plane in which have been drawn two perpendicular coordinate axes, representing respectively the real axis and the imaginary axis of $\mathbf{C}$ (Fig. 7).

As in every topological field, every rational function of $n$ complex variables with complex coefficients is continuous at every point of $\mathbf{C}^n$ at which the denominator does not vanish.

!A diagram showing the complex plane with axes labeled x and y, and a point labeled z = x + iy

Figure 7.

The permutation $z \to \overline{z}$ of $\mathbf{C}$ is continuous, and is therefore an automorphism of the topological field $\mathbf{C}$ .

In fact it is the only automorphism of the topological field $\mathbf{C}$ other than the identity automorphism (see Exercise 4).

The functions $\Re(z), \Im(z)$ are just the projections of $\mathbf{R}^2$ onto its factors, and are therefore continuous; the same is true of the absolute value $|z|$ , since it is the Euclidean norm (Chapter VI, § 2, no. 1) of the point $(x, y)$ in $\mathbf{R}^2$ .

The properties of the absolute value lead to another proof of the fact that the topology of $\mathbf{C}$ is compatible with its field structure (cf. Chapter IX, § 3, no. 2); the continuity of $z + z'$ follows from the triangle inequality $|z + z'| \leq |z| + |z'|$ ; that of $zz'$ follows from the relation

|zz' - z_0z'_0| = |z_0(z' - z'_0) + (z - z_0)z'_0 + (z - z_0)(z' - z'_0)| \\ \leq |z_0| \cdot |z' - z'_0| + |z'_0| \cdot |z - z_0| + |z - z_0| \cdot |z' - z'_0|;

lastly, the continuity of

z^{-1}

follows from the relation

|z_0^{-1} - z^{-1}| = |z|^{-1} \cdot |z - z_0| \cdot |z_0|^{-1}.

3. The multiplicative group $\mathbf{C}^*$

We know from Chapter III, § 6, no. 7 that the topology induced on the multiplicative group $\mathbf{C}^*$ of non-zero complex numbers is compatible with the group structure of $\mathbf{C}^*$ . Since $\mathbf{C}^*$ is open in $\mathbf{C}$ , it follows that $\mathbf{C}^*$ is a locally compact topological group (Chapter I, § 9, no. 7, Proposition 13) and therefore complete (with respect to the multiplicative uniformity; cf. Chapter III, § 6, no. 8, Proposition 8). The multiplicative group $\mathbf{R}_+^*$ of real numbers $> 0$ is a closed subgroup of $\mathbf{C}^*$ . Another subgroup is the set $\mathbf{U}$ of complex numbers of absolute value 1, which is identified with the unit circle $S_1$ of $\mathbf{R}^2$ , and is therefore a compact group. Moreover:

Proposition 1

The topological group $\mathbf{C}^*$ is isomorphic to the product of the topological groups $\mathbf{R}^*$ and $\mathbf{U}$ .

For the mapping $z \to \left( |z|, \frac{z}{|z|} \right)$ is a homeomorphism of $\mathbf{C}^*$ onto $\mathbf{R}_+^* \times \mathbf{U}$ (Chapter VI, § 2, no. 3, Proposition 3); and it follows immediately that it is an isomorphism of the group structures.

The topological group $\mathbf{R}_+^*$ is already known to be isomorphic to the additive group $\mathbf{R}$ (Chapter V, § 4, Theorem 1); the study of the topological group $\mathbf{C}^*$ is therefore reduced to that of $\mathbf{U}$ , which we shall consider in § 2.

4. The division ring of quaternions

It follows from Theorem 1 that the field $\mathbf{R}$ is a maximal ordered field, and therefore the only non-commutative division ring of finite rank over $\mathbf{R}$ is (up to isomorphism) the division ring of quaternions over $\mathbf{R}$ ; it is denoted by $\mathbf{H}$ and is called the division ring of real quaternions (or simply the division ring of quaternions, when there is no fear of confusion). Since $\mathbf{H}$ is of rank 4 over the field $\mathbf{R}$ , we can define a topology on $\mathbf{H}$ homeomorphic to that of $\mathbf{R}^4$ (Chapter VI, § 1, no. 5). To be precise, we shall usually identify $\mathbf{H}$ with $\mathbf{R}^4$ , the elements $i, i, j, k$ of the canonical basis of $\mathbf{H}$ being identified respectively with the vectors $e_0, e_1, e_2, e_3$ of the canonical basis of $\mathbf{R}^4$ .

We recall that the multiplication table of the canonical basis of $\mathbf{H}$ is given by the formulae

i^2 = j^2 = k^2 = -1, \quad ij = -ji = k, jk = -kj = i, \quad ki = -ik = j.

The topology of $\mathbf{H}$ is compatible not only with the ring structure of $\mathbf{H}$ (Chapter VI, § 1, no. 5) but also with its division ring structure; for if $x$ is a non-zero quaternion, the coordinates of $x^{-1}$ are rational functions of those of $x$ , whose denominators do not vanish. The division ring $\mathbf{H}$ , endowed with this topology, is therefore a non-commutative topological division ring. The quaternions $a + bi$ ( $a, b$ real) form a (topological) subfield of $\mathbf{H}$ , isomorphic to the field $\mathbf{C}$ , with which it is often identified.

We have thus a third example of a locally compact, connected topological division ring, the others being $\mathbf{R}$ and $\mathbf{C}$ . In fact these are the only topological division rings with these two properties.

We know from algebra that the reduced norm of a quaternion $x = x_0 + x_1 i + x_2 j + x_3 k$ is

N(x) = x_0^2 + x_1^2 + x_2^2 + x_3^2 = ||x||^2

(it is therefore the square of the Euclidean norm of $x$ ). Since

N(xy) = N(x)N(y),

it follows that the set of all quaternions of norm 1, which is identical with the sphere $S_3$ , forms a compact subgroup of the multiplicative group $\mathbf{H}^*$ of non-zero quaternions.

Proposition 2

The multiplicative group $\mathbf{H}^*$ of non-zero quaternions is a topological group isomorphic to the product of its subgroups $\mathbf{R}_+^*$ and $S_3$ .

Every quaternion $x / o$ can be written as $x \cdot z$ where $z$ is a quaternion with norm 1; since $\|xx'\| = \|x\|\cdot\|x'\|$ , the mapping $x \to (\|x\|, x_1, x_2)$ of $H^*$ onto $\mathbf{R}_+^* \times S_2$ is an isomorphism of the group structures, and from Chapter VI, § 2, no. 2, Proposition 2, it is a homomorphism of $H^*$ onto $\mathbf{R}_+^* \times S_2$ .

Remark 1

With the use of the relations $\|x + y\| = \|x\| + \|y\|$ and $\|xy\| = \|x\|\cdot\|y\|$ it can be proved directly, as with the field of complex numbers in no. 2, that the topology of $\mathbf{R}^4$ is compatible with the division ring structure of $H$ (cf. Chapter IX, § 3, no. 2).

Remark 2

From what has been proved it follows that the spheres $S_1$ and $S_n$ can carry a group structure compatible with their topology. We shall see later that, for each integer $n$ other than 1 and 3, there exists no group structure on $S_n$ compatible with the topology of $S_n$ .

Remark 3

Every point of the group $S_2$ has a neighbourhood homeomorphic to $\mathbf{R}^3$ (Chapter VI, § 2, no. 4, Proposition 5) but $S_4$ is not locally isomorphic to $\mathbf{R}^3$ ; for if it were it would be abelian, since it is connected (Chapter VII, § 2, no. 2, Theorem 1), and this is not the case, since $i$ and $j$ belong to $S_3$ and $ij \neq ji$ (cf. Chapter V, § 3).

Exercises

See the exercises for § 1.