8.5 Galois theory of field extensions

Galois theory connects abstract algebra and field theory, providing tools to analyze polynomial equations. It bridges the gap between algebraic structures and the solvability of equations, revolutionizing our understanding of these mathematical concepts.

At its core, Galois theory explores field extensions and their properties. It introduces key concepts like algebraic and transcendental extensions, splitting fields, and Galois groups, laying the groundwork for understanding complex algebraic relationships and equation solvability.

Fundamentals of Galois theory

• Galois theory bridges abstract algebra and field theory providing powerful tools for analyzing polynomial equations
• Developed by Évariste Galois in the 19th century revolutionized understanding of algebraic structures and solvability of equations

Field extensions basics

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• Field extension $L/K$ occurs when field $L$ contains field $K$ as a subfield
• Degree of extension $[L:K]$ measures size of $L$ relative to $K$
• Simple extensions generated by single element $L = K(α)$
• Tower law states $[L:K] = [L:M][M:K]$ for intermediate field $M$

Algebraic vs transcendental extensions

• Algebraic extensions contain elements satisfying polynomial equations over base field
• Transcendental extensions include elements not satisfying any polynomial equation
• Algebraic closure contains all roots of polynomials over a field
• Transcendence degree measures "size" of transcendental extension

Splitting fields and normal extensions

• Splitting field smallest extension where polynomial factors completely into linear terms
• Normal extensions contain all roots of any irreducible polynomial with one root in the extension
• Separable extensions have distinct roots for all irreducible polynomials
• Galois extensions combine normal and separable properties

Galois groups and correspondence

Definition of Galois group

• Galois group $Gal(L/K)$ consists of automorphisms of $L$ fixing elements of $K$
• Order of Galois group equals degree of field extension for finite Galois extensions
• Transitive group action on roots of defining polynomial
• Fixed field of Galois group precisely the base field $K$

Fundamental theorem of Galois theory

• Establishes bijective correspondence between intermediate fields and subgroups of Galois group
• Larger subgroups correspond to smaller intermediate fields
• Normal subgroups correspond to normal extensions
• Degree of intermediate extension equals index of corresponding subgroup

Lattice of intermediate fields

• Forms complete lattice under inclusion relation
• Meets correspond to field intersections
• Joins correspond to composite fields
• Dual to lattice of subgroups of Galois group under reverse inclusion

Finite fields and Galois theory

Structure of finite fields

• Exist only for prime power orders $p^n$
• Unique up to isomorphism for each order
• Multiplicative group is cyclic
• Subfields correspond to divisors of $n$

Frobenius automorphism

• Defined by $x \mapsto x^p$ in characteristic $p$
• Generates Galois group of finite field over prime subfield
• Fixed points are elements of prime subfield
• Iterates give all automorphisms of finite field

Cyclotomic polynomials

• Minimal polynomials of primitive roots of unity
• Degree equals Euler's totient function $φ(n)$
• Coefficients are integers
• Galois group isomorphic to $(Z/nZ)^*$ for $n$-th cyclotomic field

Solvability by radicals

Radical extensions

• Formed by adjoining roots of elements in base field
• Sequence of simple radical extensions
• Degree is power of prime for simple radical extension
• Compositum of radical extensions remains radical

Solvable groups

• Have subnormal series with abelian quotients
• Include all nilpotent groups (groups with upper central series reaching whole group)
• Closure properties under subgroups quotients and extensions
• Solvable Galois group necessary for solvability by radicals

Impossibility of quintic formula

• General polynomial of degree 5 or higher not solvable by radicals
• Galois group of general quintic is symmetric group $S_5$
• $S_5$ not solvable due to simplicity of alternating group $A_5$
• Specific quintics may still be solvable (cyclotomic equations)

Applications of Galois theory

Constructibility with ruler and compass

• Constructible numbers form subfield of complex numbers
• Degree of constructible extension must be power of 2
• Impossibility of angle trisection doubling cube and squaring circle
• Constructible regular $n$-gons characterized by Gauss (Fermat primes)

Insolvability of certain polynomials

• Provides rigorous proofs for impossibility of certain constructions
• Determines which regular polygons are constructible with ruler and compass
• Proves impossibility of trisecting arbitrary angles or doubling cubes with ruler and compass
• Shows certain equations like $x^5 - x - 1 = 0$ are not solvable by radicals

Galois theory in number theory

• Crucial in studying field extensions of rational numbers
• Determines Galois groups of polynomials over rationals
• Applies to solving Diophantine equations
• Connects to class field theory and reciprocity laws

Advanced concepts

Infinite Galois theory

• Extends Galois correspondence to infinite extensions
• Uses profinite groups for Galois groups
• Krull topology on Galois group
• Fundamental theorem holds with appropriate topological considerations

Inverse Galois problem

• Asks which finite groups occur as Galois groups over rationals
• Remains open in general
• Solved for solvable groups (Shafarevich)
• Connects to moduli spaces of curves and fundamental groups

Galois cohomology

• Studies cohomology of Galois groups with coefficients in Galois modules
• Applications to class field theory and arithmetic geometry
• Connects to étale cohomology and motivic cohomology
• Crucial in studying rational points on varieties

Connections to order theory

Galois connections

• Pair of order-preserving maps between partially ordered sets
• Generalizes Galois correspondence to abstract setting
• Closure operators arise from Galois connections
• Applications in formal concept analysis and rough set theory

Fixed point theorems

• Tarski's fixed point theorem for complete lattices
• Knaster-Tarski theorem for monotone functions on complete lattices
• Applications to semantics of programming languages
• Connections to modal logic and set theory

Closure operators in Galois theory

• Field closure operators (algebraic closure normal closure etc.)
• Galois closure as largest subextension with given Galois group
• Lattice of closed subsets isomorphic to lattice of subgroups
• Generalizes to closure operators in universal algebra