Proposition 2

— Let A be an algebra over K and $\Omega$ an algebraically closed extension of K. The following assertions are equivalent:

a) The algebra A is etale. b) There exists an extension of finite degree which diagonalizes A. c) The extension $\Omega$ of K diagonalizes A.

Suppose that A is etale. Let $n$ be the degree of A over K, let L be an extension of K which diagonalizes A and let $\mathscr{H}$ be the set of algebra homomorphisms of A into L. By the Cor. to Prop. 1 we have Card $\mathscr{H} = n$. On the other hand, for each $u \in \mathscr{H}$, we have $[u(A):K] \leq n$. By V, p. 18, Th. 2, the subextension L' of L generated by the images of elements of $\mathscr{H}$ is of finite degree over K. Since there exist $n$ distinct homomorphisms of A into L', the extension L' diagonalizes A, by the Cor. 1 of Prop. 1. This shows that a) implies b).

Since every extension of finite degree of K is isomorphic to a subextension of $\Omega$ (V, p. 20, Th. 1), b) implies c). Finally c) clearly implies a).