Dedekind's lemma on independence of characters

Remarks

Suppose there is a nontrivial relation, and take one $\sum a_i \chi_i = 0$ with the fewest nonzero coefficients; then at least two coefficients, say $a_1$ and $a_2$, are nonzero. Choose $h \in G$ with $\chi_1(h) \ne \chi_2(h)$. Evaluating the relation at $hg$ and subtracting $\chi_1(h)$ times the relation at $g$ gives $\sum_i a_i (\chi_i(h) - \chi_1(h)) \chi_i = 0$. This relation has fewer nonzero coefficients but a nonzero coefficient at $\chi_2$, a contradiction.

The main application concerns field homomorphisms. If $\sigma_1, \ldots, \sigma_n \colon K \to L$ are distinct field homomorphisms, their restrictions to the group $K^\times$ are distinct characters, so $\sigma_1, \ldots, \sigma_n$ are linearly independent over $L$. For example, the identity and complex conjugation on $\C$ are independent over $\C$: if $a z + b \bar z = 0$ for all $z$, then $z = 1$ gives $a + b = 0$ and $z = i$ gives $a - b = 0$, so $a = b = 0$.

The same proof works when $G$ is only a monoid and the $\chi_i$ are monoid homomorphisms into $L^\times$.

Going deeper

For a student

In Galois theory the lemma is used to show that a finite extension $L/K$ has at most $[L:K]$ automorphisms over $K$. It is the key step in the proofs of Artin's theorem on fixed fields, of Hilbert's Theorem 90 and of the normal basis theorem. It also shows that the field trace of a finite separable extension is not identically zero.