8.5 Galois theory of field extensions

Fundamentals of Galois Theory

Galois theory provides a deep link between the structure of groups and the structure of field extensions. Developed by Évariste Galois in the early 19th century, it gives a precise framework for determining when a polynomial equation can be solved by radicals. Within this course on order theory, Galois theory also serves as the motivating example for the abstract notion of a Galois connection between partially ordered sets.

Field Extensions Basics

A field extension $L/K$ is a pair of fields where $K$ sits inside $L$ as a subfield. You can think of $L$ as an enlarged version of $K$ that contains new elements.

• The degree $[L:K]$ is the dimension of $L$ as a vector space over $K$. For example, $[\mathbb{C}:\mathbb{R}] = 2$ because every complex number can be written as $a + bi$ with $a, b \in \mathbb{R}$.
• A simple extension is one generated by a single element: $L = K(\alpha)$. Here $\alpha$ is "adjoined" to $K$, and $L$ consists of all rational expressions in $\alpha$ with coefficients in $K$.
• The tower law says that if $K \subseteq M \subseteq L$, then $[L:K] = [L:M] \cdot [M:K]$. This is the field-extension analogue of the dimension formula for nested vector spaces.

Algebraic vs. Transcendental Extensions

An element $\alpha \in L$ is algebraic over $K$ if it satisfies some nonzero polynomial with coefficients in $K$. If no such polynomial exists, $\alpha$ is transcendental over $K$.

• An extension $L/K$ is algebraic if every element of $L$ is algebraic over $K$. For instance, $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ is algebraic because $\sqrt{2}$ satisfies $x^2 - 2 = 0$.
• The algebraic closure $\overline{K}$ is the smallest algebraically closed field containing $K$; it contains all roots of all polynomials over $K$.
• The transcendence degree of an extension measures how many algebraically independent transcendental elements are needed to generate it. For example, $\mathbb{R}$ over $\mathbb{Q}$ has uncountable transcendence degree.

Splitting Fields and Normal Extensions

A splitting field of a polynomial $f(x) \in K[x]$ is the smallest extension of $K$ in which $f$ factors completely into linear factors. Splitting fields are unique up to isomorphism.

• A normal extension is one where every irreducible polynomial over $K$ that has at least one root in $L$ splits completely in $L$. Equivalently, $L$ is a splitting field of some family of polynomials over $K$.
• A separable extension is one where every element's minimal polynomial has no repeated roots. Over fields of characteristic zero (like $\mathbb{Q}$ or $\mathbb{R}$), every algebraic extension is automatically separable.
• A Galois extension is one that is both normal and separable. These are the extensions where the Galois correspondence works perfectly.

Galois Groups and Correspondence

Definition of Galois Group

The Galois group $\text{Gal}(L/K)$ is the set of all field automorphisms of $L$ that fix every element of $K$, equipped with composition as the group operation.

• For a finite Galois extension, $|\text{Gal}(L/K)| = [L:K]$. So the number of symmetries equals the degree of the extension.
• The Galois group acts transitively on the roots of each irreducible polynomial over $K$ that has a root in $L$. This is what makes it so useful for studying polynomials: the group captures exactly how the roots can be permuted.
• The fixed field $L^G$ of a subgroup $G \leq \text{Gal}(L/K)$ consists of all elements left unchanged by every automorphism in $G$. For the full Galois group, $L^{\text{Gal}(L/K)} = K$.

Fundamental Theorem of Galois Theory

This is the central result. It establishes a bijective, inclusion-reversing correspondence between:

• intermediate fields $M$ with $K \subseteq M \subseteq L$, and
• subgroups $H$ of $\text{Gal}(L/K)$.

The correspondence works as follows:

1. Given an intermediate field $M$, map it to the subgroup $\text{Gal}(L/M)$ of automorphisms fixing $M$.
2. Given a subgroup $H$, map it to the fixed field $L^H$.

Key properties of this correspondence:

• It reverses inclusion: a larger intermediate field corresponds to a smaller subgroup, and vice versa.
• $[M:K] = [\text{Gal}(L/K) : \text{Gal}(L/M)]$, i.e., the degree of the sub-extension equals the index of the corresponding subgroup.
• Normal subgroups of $\text{Gal}(L/K)$ correspond exactly to intermediate fields $M$ such that $M/K$ is itself a Galois extension. In that case, $\text{Gal}(M/K) \cong \text{Gal}(L/K) / \text{Gal}(L/M)$.

Lattice of Intermediate Fields

The intermediate fields of a Galois extension $L/K$ form a complete lattice under field inclusion:

• The meet (greatest lower bound) of two intermediate fields is their intersection.
• The join (least upper bound) is their compositum (the smallest field containing both).

This lattice is dual (order-reversed) to the lattice of subgroups of $\text{Gal}(L/K)$. This duality is precisely a Galois connection in the order-theoretic sense, which is why this example is so central to Unit 8.

Finite Fields and Galois Theory

Structure of Finite Fields

Finite fields have an especially clean Galois theory because every extension of finite fields is automatically Galois.

• A finite field exists if and only if its order is a prime power $p^n$. The field of order $p^n$ is denoted $\mathbb{F}_{p^n}$ (or $GF(p^n)$).
• For each prime power, the finite field is unique up to isomorphism.
• The multiplicative group $\mathbb{F}_{p^n}^{\times}$ is cyclic of order $p^n - 1$.
• $\mathbb{F}_{p^m}$ is a subfield of $\mathbb{F}_{p^n}$ if and only if $m$ divides $n$. So the subfield lattice mirrors the divisibility lattice of the exponents.

Frobenius Automorphism

The Frobenius automorphism $\varphi: x \mapsto x^p$ is the key automorphism in characteristic $p$.

• It generates the entire Galois group $\text{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$, which is cyclic of order $n$.
• Its fixed points are exactly the elements of the prime subfield $\mathbb{F}_p$ (those satisfying $x^p = x$).
• The iterates $\varphi, \varphi^2, \ldots, \varphi^{n-1}, \varphi^n = \text{id}$ give all $n$ automorphisms of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$.

Cyclotomic Polynomials

The $n$-th cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial over $\mathbb{Q}$ of a primitive $n$-th root of unity.

• Its degree is $\varphi(n)$, Euler's totient function.
• It always has integer coefficients and is irreducible over $\mathbb{Q}$.
• The Galois group of the $n$-th cyclotomic field $\mathbb{Q}(\zeta_n)$ over $\mathbb{Q}$ is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^{\times}$, the multiplicative group of integers mod $n$. Each automorphism sends $\zeta_n \mapsto \zeta_n^k$ for some $k$ coprime to $n$.

Solvability by Radicals

Radical Extensions

A radical extension of $K$ is built by repeatedly adjoining $n$-th roots of existing elements. More precisely, it's a tower:

$K = K_0 \subset K_1 \subset \cdots \subset K_m$

where each step $K_{i+1} = K_i(\alpha_i)$ with $\alpha_i^{n_i} \in K_i$ for some positive integer $n_i$.

A polynomial $f(x) \in K[x]$ is solvable by radicals if its roots all lie in some radical extension of $K$.

Solvable Groups

A group $G$ is solvable if it has a subnormal series

$\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$

where each successive quotient $G_{i+1}/G_i$ is abelian.

• All abelian groups are solvable (trivially).
• All nilpotent groups are solvable (nilpotent is a stronger condition).
• Solvability is preserved under taking subgroups, quotients, and extensions.
• The connection to field theory: a polynomial is solvable by radicals if and only if its Galois group is a solvable group. This is the Galois criterion.

Impossibility of the Quintic Formula

The general polynomial of degree 5 or higher cannot be solved by radicals. Here's why:

1. The Galois group of the "general" degree-$n$ polynomial (one with independent indeterminate coefficients) is the symmetric group $S_n$.
2. For $n \geq 5$, the alternating group $A_n$ is simple (has no nontrivial normal subgroups).
3. Since $A_5$ is simple and non-abelian, $S_5$ has no subnormal series with abelian quotients, so $S_5$ is not solvable.
4. By the Galois criterion, the general quintic is not solvable by radicals.

Note that specific degree-5 polynomials can still be solvable. For example, $x^5 - 1 = 0$ is solved by roots of unity. The impossibility applies to a general formula covering all quintics.

Applications of Galois Theory

Constructibility with Ruler and Compass

A length is constructible (starting from a unit segment, using only straightedge and compass) if and only if it lies in a field extension of $\mathbb{Q}$ whose degree is a power of 2.

This criterion settles three famous ancient problems:

• Doubling the cube requires constructing $\sqrt[3]{2}$, which has degree 3 over $\mathbb{Q}$. Since 3 is not a power of 2, it's impossible.
• Trisecting an arbitrary angle leads to a cubic extension in general, so it's also impossible.
• Squaring the circle requires constructing $\sqrt{\pi}$, but $\pi$ is transcendental, so no algebraic extension contains it.

Gauss proved that a regular $n$-gon is constructible if and only if $n = 2^k \cdot p_1 \cdot p_2 \cdots p_m$ where the $p_i$ are distinct Fermat primes (primes of the form $2^{2^j} + 1$).

Insolvability of Certain Polynomials

Beyond the general quintic, Galois theory can determine solvability for specific polynomials. For instance, $x^5 - x - 1 = 0$ has Galois group $S_5$ over $\mathbb{Q}$, so it is not solvable by radicals. Computing the Galois group of a specific polynomial typically involves analyzing its discriminant, factorization patterns modulo primes, and resolvent polynomials.

Galois Theory in Number Theory

Galois theory is foundational in algebraic number theory:

• It determines the structure of extensions of $\mathbb{Q}$, which is central to understanding algebraic number fields.
• Decomposition and inertia groups describe how primes split in extensions.
• Class field theory classifies abelian extensions of number fields using Galois groups, connecting to reciprocity laws that generalize quadratic reciprocity.

Advanced Concepts

Infinite Galois Theory

For infinite algebraic extensions, the Galois group becomes a profinite group (an inverse limit of finite groups). The Krull topology makes it a compact, totally disconnected topological group.

The fundamental theorem still holds, but with a topological condition: the correspondence is between intermediate fields and closed subgroups of the Galois group. Open subgroups correspond to finite sub-extensions. Without the closure condition, the bijection breaks down.

Inverse Galois Problem

This asks: for every finite group $G$, does there exist a Galois extension of $\mathbb{Q}$ with Galois group isomorphic to $G$?

• The problem remains open in general.
• Shafarevich proved it for all finite solvable groups.
• It's known for many specific families: symmetric groups, alternating groups, and various simple groups.
• Approaches connect to the theory of moduli spaces of covers of curves and the arithmetic of fundamental groups.

Galois Cohomology

Galois cohomology studies the cohomology groups $H^n(G, M)$ where $G$ is a Galois group acting on a module $M$. It provides algebraic invariants that detect obstructions.

• $H^1$ classifies torsors (principal homogeneous spaces) and relates to descent problems.
• It's a key tool in class field theory and in studying rational points on algebraic varieties.
• Connections to étale cohomology and motivic cohomology place it at the heart of modern arithmetic geometry.

Connections to Order Theory

Galois Connections

The Fundamental Theorem of Galois Theory is the prototype for the abstract notion of a Galois connection. In general, a Galois connection is a pair of order-reversing maps $f: P \to Q$ and $g: Q \to P$ between two partially ordered sets satisfying $a \leq g(f(a))$ and $b \leq f(g(b))$ for all $a \in P, b \in Q$.

In the field-theoretic setting:

• $P$ is the poset of intermediate fields (ordered by inclusion).
• $Q$ is the poset of subgroups of the Galois group (ordered by inclusion).
• The maps are $M \mapsto \text{Gal}(L/M)$ and $H \mapsto L^H$.

The compositions $M \mapsto L^{\text{Gal}(L/M)}$ and $H \mapsto \text{Gal}(L/L^H)$ are closure operators. For Galois extensions, these closures are the identity (every element is closed), which is what makes the correspondence bijective.

Galois connections appear throughout mathematics and computer science, including formal concept analysis and rough set theory.

Fixed Point Theorems

Lattice-theoretic fixed point results connect to the algebraic structures in Galois theory:

• Tarski's fixed point theorem states that every order-preserving map on a complete lattice has a fixed point, and the set of all fixed points itself forms a complete lattice.
• The Knaster-Tarski theorem is the same result (the names are often used interchangeably). It applies to monotone functions on complete lattices.
• These theorems find applications in the semantics of programming languages (defining recursive types and fixed-point semantics), modal logic, and set theory.

Closure Operators in Galois Theory

Several natural closure operators arise in field theory:

• Algebraic closure: sending a field to its algebraic closure.
• Normal closure: the smallest normal extension containing a given extension.
• Galois closure: the smallest Galois extension containing a given extension.

The closed elements of the Galois connection (intermediate fields fixed by the closure operator) form a lattice isomorphic to the lattice of subgroups of the Galois group. This pattern generalizes beyond field theory to closure operators in universal algebra and formal concept analysis, tying Galois theory firmly to the order-theoretic framework of this course.