Galois Theory II: The Proof

Introduction

The proof of the Fundamental Theorem of Galois Theory relies on a series of lemmas and propositions that connect the size of the automorphism group to the degree of the extension. This chapter provides a detailed outline of the proof.

Proof Outline

Step 1: Establish Equivalence for Galois Extensions

First, one proves the equivalence of the defining properties of a finite Galois extension:

1. It is normal and separable
2. It is the splitting field of a separable polynomial
3. $|\text{Aut}(L/K)|=[L:K]$
4. The fixed field of $\text{Aut}(L/K)$ is $K$

The key lemma, due to Artin, states that for any finite group of automorphisms $H$ of a field $L$, the extension $L/L^H$ is Galois with group $H$ and degree $|H|$.

Step 2: Show the Correspondence is Bijective

With these equivalences, one shows that the maps $E\mapsto \text{Gal}(L/E)$ and $H\mapsto L^H$ are inverses of each other. The fact that $L^{\text{Gal}(L/E)}=E$ follows because $L/E$ is itself a Galois extension. The fact that $\text{Gal}(L/L^H)=H$ is a direct consequence of Artin's lemma.

Step 3: Prove the Properties

The properties of the correspondence (inclusion-reversal, degrees/indices, and normality) are then proven using the established bijection and the definitions of degree, index, and normal subgroups.

Key Lemmas

Artin's Lemma

Lemma 18.1 (Artin's Lemma): Let $L$ be a field and let $H$ be a finite group of automorphisms of $L$. Let $K=L^H$ be the fixed field of $H$. Then:

1. $L/K$ is a Galois extension
2. $\text{Gal}(L/K)=H$
3. $[L:K]=|H|$

Proof of Artin's Lemma

The proof involves several steps:

1. Show that $L/K$ is algebraic: Every element of $L$ is algebraic over $K$
2. Show that $L/K$ is separable: The minimal polynomial of any element has distinct roots
3. Show that $L/K$ is normal: Every irreducible polynomial with a root in $L$ splits in $L$
4. Show that $H=\text{Gal}(L/K)$: Every element of $H$ is a $K$-automorphism, and every $K$-automorphism is in $H$

Dedekind's Lemma

Lemma 18.2 (Dedekind's Lemma): Let $L$ be a field and let ${\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n$ be distinct automorphisms of $L$. Then ${\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n$ are linearly independent over $L$.

This lemma is crucial for showing that the number of automorphisms cannot exceed the degree of the extension.

The Correspondence

Definition of the Maps

Let $L/K$ be a finite Galois extension with Galois group $G=\text{Gal}(L/K)$. We define two maps:

1. Field to Group: For each intermediate field $E$ (where $K\subseteq E\subseteq L$), we associate the subgroup of automorphisms that fix $E$: $H=\text{Gal}(L/E)$
2. Group to Field: For each subgroup $H\le G$, we associate its fixed field: $E=L^H=\{x\in L\mid \sigma (x)=x\text{ for all }\sigma \in H\}$

Bijectivity

Theorem 18.3: The maps $E\mapsto \text{Gal}(L/E)$ and $H\mapsto L^H$ are inverses of each other.

Proof:

• For any intermediate field $E$, we have $L^{\text{Gal}(L/E)}=E$ because $L/E$ is a Galois extension
• For any subgroup $H\le G$, we have $\text{Gal}(L/L^H)=H$ by Artin's lemma

Properties of the Correspondence

Inclusion-Reversing

Theorem 18.4: The correspondence is inclusion-reversing:

• If $E_1\subseteq E_2$, then $\text{Gal}(L/E_2)\le \text{Gal}(L/E_1)$
• Conversely, if $H_1\le H_2$, then $L^{H_2}\subseteq L^{H_1}$

Degrees and Indices

Theorem 18.5: For any intermediate field $E$ and corresponding subgroup $H=\text{Gal}(L/E)$:

• $[L:E]=|H|$
• $[E:K]=[G:H]$

Normality

Theorem 18.6: The extension $E/K$ is a normal (and thus Galois) extension if and only if its corresponding subgroup $H=\text{Gal}(L/E)$ is a normal subgroup of $G$. In this case, the Galois group of $E/K$ is isomorphic to the quotient group $G/H$:

Applications of the Proof

Application 1: Understanding Field Structure

The proof shows how the structure of the Galois group reflects the structure of the field extension, providing a powerful tool for understanding field theory.

Application 2: Computing Galois Groups

The correspondence allows us to compute Galois groups by understanding the intermediate fields and their relationships.

Application 3: Solvability by Radicals

The proof provides the foundation for understanding when polynomial equations can be solved by radicals, as the structure of the Galois group determines the solvability.

Examples

Example 1: Quadratic Extensions

For the extension $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$:

• The Galois group is $\{id,\sigma \}$ where $\sigma (\sqrt{2})=-\sqrt{2}$
• The only intermediate field is $\mathbb{Q}$ itself
• The correspondence is: $\mathbb{Q}\leftrightarrow \{id,\sigma \}$

Example 2: Cubic Extensions

For the extension $\mathbb{Q}(\sqrt[3]{2},\omega )/\mathbb{Q}$:

• The Galois group is $S_3$ (the symmetric group on 3 letters)
• The intermediate fields correspond to the subgroups of $S_3$
• The normal subgroups correspond to normal extensions

Example 3: Cyclotomic Extensions

For the extension $\mathbb{Q}({\zeta }_n)/\mathbb{Q}$ where ${\zeta }_n$ is a primitive $n$-th root of unity:

• The Galois group is isomorphic to $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$
• The intermediate fields correspond to the subgroups of this group
• All intermediate fields are normal over $\mathbb{Q}$

Summary

The proof of the Fundamental Theorem of Galois Theory is a beautiful synthesis of field theory and group theory. It establishes a precise correspondence between the structure of field extensions and the structure of groups, providing a powerful tool for understanding both subjects.

The key insight is that the automorphisms of a field extension form a group that encodes the structure of the extension, and this group-theoretic information can be used to understand the field-theoretic properties.

This correspondence is fundamental to modern algebra and has applications throughout mathematics, from number theory to algebraic geometry to cryptography.