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Artin's fixed-field lower bound $[K : K^{G}] \geq ∣ G ∣$

Statement

If $G$ is a finite group of automorphisms of $K$ , then $[K : K^{G}] \geq ∣ G ∣$ .

Facts & Assumptions

Given: A field $K$ , a finite group $G = {σ_{1}, \dots, σ_{m}}$ of automorphisms of $K$ , and its fixed field $K^{G}$ (The fixed field $K^{G}$ of a group of field automorphisms).

[L1]

Every finite family of distinct group homomorphisms $G \to K^{\times}$ is linearly independent over $K$ as a family of functions (Dedekind's linear independence theorem for distinct characters).

Proof

technique · direct

1.1

Restricted to $K^{\times}$ , the distinct automorphisms $σ_{1}, \dots, σ_{m}$ are distinct characters $K^{\times} \to K^{\times}$ , so [L1] makes them linearly independent as functions. Hence their evaluation vectors span $K^{m}$ : otherwise a nonzero linear functional on their span would give a nontrivial $K$ -linear relation among the $σ_{i}$ . Choose nonzero $x_{1}, \dots, x_{m} \in K$ such that the evaluation matrix $A = (σ_{i} (x_{j}))_{i, j}$ is invertible. For $m = 1$ , one may take $x_{1} = 1$ .

L1choose

2.1

Suppose $c_{1}, \dots, c_{m} \in K^{G}$ satisfy $\sum_{j} c_{j} x_{j} = 0$ . Applying each $σ_{i}$ and using $σ_{i} (c_{j}) = c_{j}$ gives $A (c_{1}, \dots, c_{m})^{T} = 0$ , so invertibility of $A$ forces every $c_{j} = 0$ . Thus $x_{1}, \dots, x_{m}$ are linearly independent over $K^{G}$ .

step 1.1L1algebra

3.1

A $K^{G}$ -linearly independent family of $m = ∣ G ∣$ elements of $K$ gives $[K : K^{G}] \geq m = ∣ G ∣$ .

step 2.1∎

Depends on

Used by

Artin's fixed-field theorem: [K:K^G]=|G| and Aut(K/K^G)=G Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Fields and Galois Theory, v5.10, Theorem 3.4 (standard reference, not scraped)
K. Conrad, The Galois Correspondence, Section 5 (standard reference, not scraped)

Artin's fixed-field lower bound [K:KG]≥∣G∣[K:KG]≥∣G∣