Exercise 35, § 2

Let A be an algebra over $\mathbf{R}$ which is a field, endowed with a separated locally convex topology such that the multiplication in A is continuous, and such that the mapping $x\mapsto x^{-1}$ is continuous at 1. Then A is $\mathbf{R}$-isomorphic either to $\mathbf{R}$, or to $\mathbf{C}$, or to the field of quaternions $\mathbf{H}$. (If A is commutative and there exists $u\in A$ with $u^2=-1$, endow A with a structure of algebra over $\mathbf{C}$, and apply Exercise 34, f).) If A is commutative and there does not exist an element $u\in A$ such that $u^2=-1$, apply Exercise 34, f ) to the field $A\otimes_{\mathbf{R}}\mathbf{C}$. If A is noncommutative, reason as in AC, VI, §6, n$^o4$, Theorem 1, third case.)