3.4 Applications of Galois theory to number fields

Galois theory bridges number fields and group theory, unveiling deep connections between field extensions and their automorphisms. It provides a powerful framework for analyzing polynomial equations, field structures, and algebraic properties of number fields.

The fundamental theorem of Galois theory establishes a bijective correspondence between intermediate fields and subgroups of the Galois group. This correspondence allows us to tackle problems in number theory, including solvability of equations and properties of algebraic extensions.

Galois theory for number fields

Fundamental concepts and correspondences

• Galois theory establishes a correspondence between field extensions and group theory providing a powerful framework for analyzing number field structures
• Galois group of a field extension K/F represents the group of automorphisms of K fixing every element of F
• Fundamental theorem of Galois theory creates a bijective correspondence between intermediate fields of a Galois extension and subgroups of its Galois group
• Normal closure of a number field represents the smallest Galois extension containing the original field
• Splitting fields of polynomials over number fields play a crucial role in understanding field extension structures
• Discriminant of a polynomial provides information about the nature of its roots and splitting field structure

Important tools and properties

• Norm and trace of elements in number field extensions serve as important tools for studying algebraic properties
• Field extension degree equals the order of the Galois group for Galois extensions
• Transitive subgroups of symmetric groups hold particular importance in determining Galois groups of irreducible polynomials
• Polynomial discriminant offers insights into the Galois group structure, particularly regarding even and odd permutations
• Factorization patterns of defining polynomials modulo various primes help narrow down Galois group possibilities
• Advanced techniques for determining Galois groups of higher-degree extensions include resolvent polynomials and Frobenius elements

Galois groups of field extensions

Computing Galois groups

• Determine Galois group of a number field extension by analyzing automorphism actions on defining polynomial roots
• Symmetric and alternating groups frequently appear as Galois groups of number field extensions, especially in low-degree cases ($S_3$, $A_4$)
• Examine polynomial factorization patterns modulo various primes to narrow down Galois group possibilities
• Apply advanced techniques such as resolvent polynomials and Frobenius elements for higher-degree extensions
• Utilize computational algebra systems (GAP, Magma) to assist in Galois group calculations for complex cases

Properties and applications

• Galois group order equals the field extension degree for Galois extensions
• Transitive subgroups of symmetric groups hold particular importance for Galois groups of irreducible polynomials
• Polynomial discriminant provides information about Galois group structure, especially regarding even and odd permutations
• Galois group structure reveals important properties of the field extension (normal, separable, radical)
• Apply Galois groups to solve problems in number theory (class field theory, reciprocity laws)

Solvability of polynomial equations

Solvability criteria and theorems

• Solvability by radicals directly relates to the Galois group structure of the polynomial's splitting field
• Polynomial equation solvability by radicals occurs if and only if its Galois group represents a solvable group
• Abel-Ruffini theorem states the non-existence of general algebraic solutions for polynomial equations of degree five or higher
• Galois' criterion for solvability provides a group-theoretic characterization of solvable polynomials
• Composition series and derived series of a group serve as essential tools for determining Galois group solvability

Radical extensions and examples

• Radical extensions and their connection to cyclic extensions play a crucial role in understanding solvability by radicals
• Analyze specific examples of solvable and unsolvable quintic polynomials to illustrate Galois theory application ($x^5 - 4x + 2$, unsolvable)
• Explore solvable polynomials of degree less than 5 and their solution methods (quadratic formula, cubic formula)
• Investigate the structure of radical extensions and their Galois groups ($\mathbb{Q}(\sqrt[n]{a})$ over $\mathbb{Q}$)
• Study the Galois groups of cyclotomic extensions and their connection to solvability ($\mathbb{Q}(\zeta_n)$ over $\mathbb{Q}$, where $\zeta_n$ is a primitive nth root of unity)

Galois theory vs Fundamental Theorem of Algebra

Fundamental Theorem of Algebra and Galois theory

• Fundamental Theorem of Algebra states every non-constant polynomial with complex coefficients has at least one complex root
• Galois theory provides an elegant proof of the Fundamental Theorem of Algebra by considering the Galois group of $\mathbb{C}$ over $\mathbb{R}$
• Algebraic closure of a field represents the smallest field extension containing all roots of polynomials with coefficients in the original field
• Galois group of $\mathbb{C}$ over $\mathbb{R}$ isomorphic to the cyclic group of order 2, reflecting the complex conjugation automorphism
• Fixed field of the Galois group of $\mathbb{C}$ over $\mathbb{R}$ precisely equals $\mathbb{R}$, illustrating the Galois correspondence

Generalizations and connections

• Fundamental Theorem of Algebra generalizes to show $\mathbb{C}$ algebraically closed, meaning every non-constant polynomial splits completely over $\mathbb{C}$
• Connection between Galois theory and Fundamental Theorem of Algebra highlights interplay between algebraic and topological properties of complex numbers
• Explore the Galois groups of finite extensions of $\mathbb{Q}$ contained in $\mathbb{C}$ (cyclotomic fields, quadratic extensions)
• Investigate the Galois theory of infinite algebraic extensions ($\overline{\mathbb{Q}}$ over $\mathbb{Q}$)
• Analyze the connection between Galois theory and the theory of Riemann surfaces in complex analysis