5.1 Galois groups

Galois groups link field theory and group theory, capturing symmetries of field extensions. They're key to understanding algebraic structures and analyzing polynomial equations, providing insights into the nature of solutions.

In arithmetic geometry, Galois groups help study geometric objects over various fields. They connect number theory, algebra, and geometry, allowing us to analyze deep arithmetic properties of curves, varieties, and other mathematical structures.

Definition of Galois groups

• Galois groups form a fundamental concept in arithmetic geometry connecting field theory and group theory
• These groups capture the symmetries of field extensions crucial for understanding algebraic structures
• Galois groups provide a powerful tool for analyzing polynomial equations and their solutions

Field extensions

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• Extend a base field F by adjoining elements to create a larger field E
• Degree of extension [E:F] measures the dimension of E as a vector space over F
• Algebraic extensions involve adjoining roots of polynomials ($\mathbb{Q}(\sqrt{2})$)
• Transcendental extensions add elements not algebraic over the base field ($\mathbb{Q}(π)$)

Automorphisms of fields

• Field automorphisms preserve addition and multiplication operations
• Form a group under composition, key to defining Galois groups
• Identity automorphism maps every element to itself
• Non-trivial automorphisms permute roots of irreducible polynomials
  • $x^2 - 2$ has automorphism swapping $\sqrt{2}$ and $-\sqrt{2}$

Fixed fields

• Elements left unchanged by all automorphisms in a given group
• Fundamental in establishing the Galois correspondence
• Galois group of E/F fixes precisely the elements of F
• Degree of extension equals order of Galois group for Galois extensions

Properties of Galois groups

• Galois groups encapsulate the structure and symmetries of field extensions
• These properties play a crucial role in solving polynomial equations
• Understanding Galois group properties aids in analyzing arithmetic geometric objects

Group structure

• Galois groups can be cyclic, abelian, solvable, or more complex
• Determined by the nature of the field extension and its generating polynomial
• Symmetric groups S_n often appear as Galois groups of general polynomials
• Direct and semidirect products arise from composite field extensions

Order of Galois groups

• Equals the degree of the field extension for Galois extensions
• Divides the degree for non-Galois extensions
• Prime degree extensions have cyclic Galois groups
• Order determines the complexity of the extension and solvability of equations

Transitive vs intransitive groups

• Transitive Galois groups act transitively on the roots of the defining polynomial
• Correspond to irreducible polynomials over the base field
• Intransitive groups indicate the polynomial factors over the base field
• Imprimitive transitive groups suggest the existence of intermediate fields

Fundamental theorem of Galois theory

• Establishes a profound connection between subfields and subgroups
• Forms the cornerstone of Galois theory in arithmetic geometry
• Provides a powerful tool for analyzing field extensions and their properties

Galois correspondence

• Bijection between intermediate fields and subgroups of the Galois group
• Larger subgroups correspond to smaller intermediate fields
• Preserves inclusion relations between subgroups and subfields
• Allows translation of field-theoretic problems into group-theoretic ones

Subgroups and subfields

• Each subgroup H of Gal(E/F) corresponds to a unique intermediate field K
• Fixed field of H equals the corresponding intermediate field K
• [E:K] equals the order of H
• Normal subgroups correspond to normal extensions

Normal subgroups vs normal extensions

• Normal subgroups correspond to normal (Galois) extensions
• Normal extensions have all conjugates of any root in the extension
• Quotient group Gal(E/F)/Gal(E/K) isomorphic to Gal(K/F) for normal K/F
• Simplifies the study of tower extensions in arithmetic geometry

Finite fields and Galois groups

• Finite fields play a crucial role in arithmetic geometry and cryptography
• Galois groups of finite fields have a particularly nice structure
• Understanding these groups aids in analyzing curves over finite fields

Frobenius automorphism

• Generates the Galois group of finite field extensions
• Maps x to x^q where q is the order of the base field
• Order equals the degree of the extension
• Crucial in studying zeta functions of varieties over finite fields

Cyclotomic fields

• Extensions of Q obtained by adjoining roots of unity
• Galois group isomorphic to (Z/nZ)* for the nth cyclotomic field
• Important in studying elliptic curves and modular forms
• Cyclotomic extensions are abelian, key to class field theory

Algebraic closure

• Contains all roots of polynomials with coefficients in the field
• Finite fields have unique algebraic closures up to isomorphism
• Galois group of algebraic closure over F_q is profinite completion of Z
• Essential in defining l-adic representations in arithmetic geometry

Solvability and Galois groups

• Solvability of polynomial equations connects deeply to Galois group structure
• This concept bridges abstract algebra and classical problems in mathematics
• Understanding solvability aids in analyzing arithmetic geometric objects

Solvable groups

• Groups with a normal series where each quotient is abelian
• Characterized by a tower of abelian extensions in field theory
• Include all abelian groups, dihedral groups, and symmetric groups S_n for n ≤ 4
• Solvable Galois groups correspond to equations solvable by radicals

Radical extensions

• Field extensions obtained by adjoining nth roots
• Galois groups of pure radical extensions are always abelian
• General radical extensions have solvable Galois groups
• Key to understanding which polynomial equations are solvable by radicals

Insolvability of quintic equations

• General quintic equations not solvable by radicals
• Galois group S_5 of general quintic is not solvable
• Abel-Ruffini theorem proves this insolvability
• Motivates the study of transcendental methods for solving equations

Applications in arithmetic geometry

• Galois theory provides powerful tools for studying geometric objects
• These applications connect number theory, algebra, and geometry
• Understanding Galois groups aids in analyzing deep arithmetic properties

Elliptic curves

• Galois representations arise from torsion points on elliptic curves
• Galois groups of division fields crucial in studying rational points
• Weil pairing connects Galois representations to modular forms
• Serre's open image theorem relates to Galois representations of elliptic curves

Galois representations

• Continuous homomorphisms from absolute Galois groups to matrix groups
• Arise naturally from cohomology of varieties
• l-adic representations crucial in studying arithmetic of varieties
• Connect Galois theory to automorphic forms via Langlands program

L-functions and Galois groups

• Encode deep arithmetic information about varieties
• Galois groups determine local factors of L-functions
• Artin L-functions associated to Galois representations
• Sato-Tate conjecture relates Galois groups to distribution of Frobenius elements

Computational aspects

• Efficient algorithms for Galois groups essential in arithmetic geometry
• Computational methods allow exploration of complex Galois-theoretic problems
• These tools bridge theoretical understanding and practical applications

Algorithms for Galois groups

• Polynomial factorization over various fields (finite fields, number fields)
• Resolvent methods for computing Galois groups
• Stauduhar's algorithm for transitive permutation groups
• Pohst-Zassenhaus algorithm for splitting fields and Galois groups

Software tools

• Computer algebra systems (Magma, SageMath, GAP)
• Specialized Galois theory packages (PARI/GP, Macaulay2)
• Databases of Galois groups and number fields (LMFDB)
• Visualization tools for Galois group lattices and subfield structures

Complexity considerations

• Polynomial-time algorithms exist for Galois groups over finite fields
• Exponential-time algorithms generally required for number fields
• Space complexity important for storing large Galois groups
• Probabilistic algorithms often more efficient than deterministic ones

Advanced topics

• These areas represent current research frontiers in Galois theory
• Understanding these topics aids in tackling open problems in arithmetic geometry
• Advanced Galois theory connects to deep results in various mathematical fields

Inverse Galois problem

• Asks which finite groups occur as Galois groups over a given field
• Solved for algebraically closed fields and finite fields
• Open problem for Q, partial results known (all solvable groups realized)
• Connects to moduli spaces of curves and fundamental groups in geometry

Infinite Galois theory

• Studies Galois groups of infinite extensions
• Topological groups replace finite groups in this setting
• Profinite completion of groups plays a crucial role
• Applications to class field theory and anabelian geometry

Galois cohomology

• Cohomology of Galois groups with coefficients in various modules
• Connects Galois theory to homological algebra and algebraic K-theory
• Crucial in studying arithmetic of elliptic curves (Selmer groups)
• Tate cohomology relates to special values of L-functions