---
tag: N43G50YHG2
kind: statement
role: lemma
name: Dedekind's lemma on independence of characters
msc: [12F10]
uses: [0HV9Y7WG5R, 05THZKR9AN, 0EVHJFTG9Z, EZVN87C88J]
about: [0CRMGPAH83]
sources:
  - url: https://en.wikipedia.org/wiki/Character_(mathematics)
---

## Statement

Let $G$ be a group and let $L$ be a field, with multiplicative [[group of units|05THZKR9AN]] $L^\times$.
Let $\chi_1, \ldots, \chi_n \colon G \to L^\times$ be pairwise distinct [[group homomorphisms|0HV9Y7WG5R]].
Then $\chi_1, \ldots, \chi_n$ are [[linearly independent|0EVHJFTG9Z]] over $L$ as functions $G \to L$: if $a_1, \ldots, a_n \in L$ and $a_1 \chi_1(g) + \cdots + a_n \chi_n(g) = 0$ for all $g \in G$, then $a_1 = \cdots = a_n = 0$.

## 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|EZVN87C88J]].
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 $L^\times$.

## Explanations

### L1 human

A character is a map from a group to the nonzero numbers of a field that turns products into products.
Dedekind's lemma says that different characters are genuinely different as functions: no one of them is a combination of the others.
It is the field-theoretic cousin of the fact that the functions $e^{a_1 x}, \ldots, e^{a_n x}$ with distinct $a_i$ are linearly independent.

### L2 human

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|X7R15C00ET]], of [[Hilbert's Theorem 90|YFF8B5A0QG]] and of the [[normal basis theorem|PXY1WYNRD4]].
It also shows that the [[field trace|WPE5Z53QYJ]] of a finite separable extension is not identically zero.