Lemma 1

Let D be a field with centre L such that, for all $x \in D, L(x)$ is an extension of L of degree $\leq m$. Then the rank of D over L is $\leq m^2$.

We may obviously restrict our attention to the case where $D \neq L$. Then there exists in D a finite separable algebraic commutative extension E of L of degree $> 1$ (Algebra, Chapter VIII, § 10, no. 3, Lemma 1); as $E = L(x)$ for some suitable x in E (Algebra, Chapter V, § 7, no. 7, Proposition 12 and Chapter VII, § 5, no. 7), by hypothesis $[E : L] \leq m$. Suppose the separable extension E is taken such that $[E : L]$ is finite and as great as possible and consider the centralizer $E' \supset E$ of E in D, which is a field of centre E such that

(Algebra, Chapter VIII, § 10, no. 2, Theorem 2). If E $\#$ E', there would exist in $\mathbf{E}'$ a finite separable algebraic extension F of $\mathbf{E}$ of degree $> 1$ (Algebra, Chapter VIII, § 10, no. 3, Lemma 1); F would therefore be a finite separable algebraic extension of L (Algebra, Chapter V, § 7, no. 4, Proposition 7) of degree $> [\mathrm{E}: \mathrm{L}]$, contrary to the definition of E; therefore $\mathbf{E}' = \mathbf{E}$, whence $[\mathrm{D}: \mathrm{L}] = [\mathrm{D}: \mathrm{E}][\mathrm{E}: \mathrm{L}] \leq m^2$.

Applying this lemma to K with $m = 2$, it is seen that K is a non-commutative extension field of R of finite rank and hence isomorphic to the field of quaternions $\mathbf{H}$ (Algebra, Chapter VIII, § 11, no. 2, Theorem 2).

Remark (1) We shall give in the chapter devoted to normed algebras a shorter proof of the Gelfand-Mazur Theorem which is valid for every Hausdorff locally convex topological algebra K over R and whose principle is the following: it is reduced (as in cases (B) and (C)) to the case where K is a commutative algebra over $\mathbf{C}$; if $x \in \mathbf{K} - \mathbf{C}.1$, we consider as above the mapping $z \mapsto (x - z.1)^{-1}$ of $\mathbf{C}$ to K, which is continuous and differentiable on $\mathbf{C}$. For every element $x'$ of the dual $\mathbf{K}'$ of the locally convex space K, $z \mapsto \langle (x - z.1)^{-1}, x' \rangle$ is then a bounded integral function on $\mathbf{C}$ and therefore constant by Liouville's Theorem and we conclude as in part (A) of the proof of Theorem 1 that this necessarily implies $\langle (x - z.1)^{-1}, x' \rangle = 0$ for all $z \in \mathbf{C}$ and all $x' \in \mathbf{K}'$; the Hahn-Banach Theorem shows that this conclusion is absurd, since $(x - z.1)^{-1} \neq 0$. Note that the argument in part (A) of the proof of Theorem 1 differs from the above only in appearance, for this argument is only a special case of that which serves to prove the maximum principle for analytic functions, the summation over the roots of unity and and passing to the limit being equivalent to calculating the integral $\int_{\gamma} \frac{\mathbf{F}(z + t)}{t} dt$ along a circle of centre 0 and the use of Cauchy's formula being avoided here, thanks to the particular form of the function F.