Theorem 1

— Let L be an extension of a field K and A an algebra over K; let $\mathscr{H}$ be the set of all K-algebra homomorphisms of A into L. Then $\mathscr{H}$ is a free subset of the vector space $\mathrm{Hom}_K(A, L)$ over L.

Let us show by induction on the integer $n \geq 0$ that every sequence $(u_1, \ldots, u_n)$ of distinct elements of $\mathscr{H}$ is free. The case $n = 0$ being trivial, we may henceforth suppose that $n \geq 1$; let $a_1, \ldots, a_n$ be elements of L such that

For $x,\ y$ in A we have whence By the induction hypothesis, the elements $u_1, \ldots, u_{n-1}$ of $\mathscr{H}$ are linearly independent, whence $\alpha_i [u_i(x) - u_n(x)] = 0$ for $1 \leq i \leq n-1$ and for all $x$ in A. Since the $u_i$ are distinct, this implies that $\alpha_i = 0$ for $i \neq n$, hence $\alpha_n u_n = 0$ and so $\alpha_n = \alpha_n u_n(1) = 0$ (on denoting by 1 the unit element of A). We have thus shown that $a_1, \ldots, a_{n-1}, \alpha_n$ are zero, and this proves the theorem.