Dedekind Independence Lemma

Lemma (Dedekind independence).
Let $L$ be a field and $\sigma_1,\dots,\sigma_m: L\to \Omega$ be distinct field homomorphisms into a commutative ring (or field) $\Omega$. If

with coefficients $a_i\in \Omega$, then $a_1=\cdots=a_m=0$.
Equivalently, $\{\sigma_1,\dots,\sigma_m\}$ is linearly independent over $\Omega$ inside the $\Omega$-module of functions $L\to\Omega$.

This is a key input for Artin's theorem on fixed fields .

Examples.

1. Over $L=\mathbb{C}$, the maps $\mathrm{id}$ and complex conjugation $\overline{\phantom{x}}$ are distinct, hence linearly independent as $\mathbb{C}$-valued functions on $\mathbb{C}$.
2. If $L/K$ is Galois and $G=\mathrm{Gal}(L/K)$, then the distinct $\sigma\in G$ are linearly independent. In particular, a relation $\sum_{\sigma\in G}a_\sigma\sigma=0$ forces all $a_\sigma=0$.
3. For $L=\mathbb{F}_{p^n}$, the maps $x\mapsto x^{p^i}$ ($0\le i<n$) are distinct automorphisms, hence independent.

Related. field automorphisms , trace .