🏃🏽‍♀️Galois Theory Unit 7 – Galois Groups and Galois Extensions

Key Concepts and Definitions

• Galois theory studies the relationship between field extensions and group theory
• A field extension $L/K$ consists of a base field $K$ and a larger field $L$ containing $K$
• An automorphism of a field extension $L/K$ is a bijective homomorphism from $L$ to itself that fixes $K$
• The Galois group $\text{Gal}(L/K)$ is the group of all automorphisms of $L/K$ under function composition
• A field extension $L/K$ is Galois if it is both normal (every irreducible polynomial in $K[x]$ that has a root in $L$ splits completely in $L[x]$) and separable (every element of $L$ is separable over $K$)
  • An element $\alpha \in L$ is separable over $K$ if its minimal polynomial over $K$ has distinct roots in an algebraic closure of $K$
• The Galois correspondence establishes a one-to-one correspondence between intermediate fields of a Galois extension and subgroups of its Galois group

Historical Context and Motivation

• Galois theory originated from Évariste Galois' work on solving polynomial equations by radicals in the early 19th century
• The unsolvability of the general quintic equation by radicals was a major motivation for developing Galois theory
• Galois introduced the concept of a "group" of permutations and connected it to the study of field extensions
• The Galois correspondence provided a powerful tool for understanding the structure of field extensions and their relationship to group theory
• Galois theory has since found numerous applications in various branches of mathematics, including algebraic geometry, number theory, and cryptography

Field Extensions: The Basics

• A field extension $L/K$ is a field $L$ containing a subfield $K$
• The degree $[L:K]$ of a field extension is the dimension of $L$ as a vector space over $K$
• A field extension $L/K$ is finite if $[L:K]$ is finite, otherwise it is infinite
• An algebraic extension is a field extension where every element of $L$ is algebraic over $K$ (i.e., a root of some polynomial in $K[x]$)
  • A transcendental extension is a field extension that is not algebraic
• A simple extension $K(\alpha)$ is obtained by adjoining a single element $\alpha$ to $K$
• The splitting field of a polynomial $f(x) \in K[x]$ is the smallest field extension of $K$ in which $f(x)$ factors into linear factors

Galois Groups: Structure and Properties

• The Galois group $\text{Gal}(L/K)$ is the group of all $K$-automorphisms of $L$ under function composition
• For a Galois extension $L/K$, the order of $\text{Gal}(L/K)$ equals the degree $[L:K]$
• The Galois group acts on the roots of polynomials in $K[x]$ that split in $L$
  • This action is transitive if the polynomial is irreducible over $K$
• The fixed field of a subgroup $H \leq \text{Gal}(L/K)$ is the set of elements in $L$ fixed by every automorphism in $H$
• The Galois group of a polynomial $f(x) \in K[x]$ is the Galois group of its splitting field over $K$
• The Fundamental Theorem of Galois Theory states that for a Galois extension $L/K$, there is a one-to-one correspondence between intermediate fields and subgroups of $\text{Gal}(L/K)$

Galois Correspondence

• The Galois correspondence is a one-to-one correspondence between intermediate fields of a Galois extension $L/K$ and subgroups of its Galois group $\text{Gal}(L/K)$
• For each intermediate field $M$ with $K \subseteq M \subseteq L$, there is a corresponding subgroup $\text{Gal}(L/M) \leq \text{Gal}(L/K)$
• Conversely, for each subgroup $H \leq \text{Gal}(L/K)$, there is a corresponding intermediate field $L^H$ (the fixed field of $H$)
• The Galois correspondence reverses inclusions: if $M_1 \subseteq M_2$ are intermediate fields, then $\text{Gal}(L/M_2) \leq \text{Gal}(L/M_1)$
• The degree of an intermediate field $M$ equals the index of its corresponding subgroup: $[L:M] = [\text{Gal}(L/K) : \text{Gal}(L/M)]$
• The Galois correspondence allows us to study field extensions by examining the structure of their Galois groups

Solvability and Radical Extensions

• A polynomial $f(x) \in K[x]$ is solvable by radicals if its roots can be expressed using the four arithmetic operations and taking $n$-th roots (radicals)
• An extension $L/K$ is a radical extension if $L$ can be obtained from $K$ by successively adjoining roots of elements in $K$
• A group $G$ is solvable if it has a subnormal series with abelian factors
  • That is, there exist subgroups $1 = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G$ such that each quotient $G_{i+1}/G_i$ is abelian
• A polynomial $f(x) \in K[x]$ is solvable by radicals if and only if its Galois group over $K$ is solvable
• The Abel-Ruffini Theorem states that for $n \geq 5$, the general polynomial of degree $n$ is not solvable by radicals
• The Galois group of a radical extension is always solvable

Applications and Examples

• Galois theory can be used to prove the impossibility of certain geometric constructions with compass and straightedge (doubling the cube, trisecting an angle, squaring the circle)
• The Galois group of the splitting field of $x^n - 1$ over $\mathbb{Q}$ is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^\times$, the multiplicative group of integers modulo $n$
• The Galois group of the splitting field of $x^p - 2$ over $\mathbb{Q}$, where $p$ is an odd prime, is isomorphic to the cyclic group $C_p$
• Galois theory has applications in algebraic number theory, such as studying the Galois group of the maximal abelian extension of $\mathbb{Q}$ (the Kronecker-Weber Theorem)
• Error-correcting codes and cryptographic protocols often rely on properties of Galois fields (finite fields)

Common Pitfalls and Tips

• Remember that not every field extension is Galois; always check for normality and separability
• Be careful when computing Galois groups: the order of the Galois group equals the degree of the extension, but not every permutation of the roots necessarily corresponds to an automorphism
• Use the Galois correspondence to relate intermediate fields and subgroups of the Galois group
• To determine solvability by radicals, check if the Galois group is a solvable group
• When working with finite fields, be mindful of the characteristic and the Frobenius automorphism
• Galois theory is a powerful tool, but it may not always be the most efficient approach for solving specific problems; consider alternative methods when appropriate