Corollary

Let L be an extension of K and $\mathscr{H}$ the set of algebra homomorphisms of A into L. We have Card $\mathscr{H} \leq [A : K]$, with equality if and only if A is diagonalized by L. If A is diagonalized by L, then $\mathscr{H}$ is a basis of the vector L-space $\mathrm{Hom}_K(A, L)$.

The vector space $\mathrm{Hom}_K(A, L)$ over L has dimension $[A : K]$, by Formula (2), and $\mathscr{H}$ is a free subset of $\mathrm{Hom}_K(A, L)$ by Th. 1 (V, p. 27). We thus have Card $\mathscr{H} \leq [A : K]$ with equality if and only if $\mathscr{H}$ is a basis of $\mathrm{Hom}_K(A, L)$. There exists an isomorphism of vector L-spaces, say $\pi : \mathrm{Hom}_K(A, L) \to A_{(L)}^*$, characterized by $u(x) = (\pi u)(1 \otimes x)$ for $x \in A$, and $\pi$ maps $\mathscr{H}$ onto the set $\mathscr{H}_L$ of L-algebra homomorphisms of $A(\cdot)$ into L. Finally the equivalence of a) and c) in Prop. 1 shows that the algebra $A_{(L)}$ over L is diagonalizable if and only if $\mathscr{H}_L$ generates the vector space $A_{(L)}^*$ over L. This completes the proof of the Corollary.