Mathematics Foundations/7.5 Galois Theory

7.5 Galois Theory 7.5.1 Introduction to Galois Theory Galois theory, named after the French mathematician Évariste Galois (1811-1832), establishes a profound connection between field theory and group theory. It provides a framework for determining whether a polynomial equation is solvable by radicals—that is, whether its solutions can be expressed using arithmetic operations and root extractions. Historical Context The theory emerged from the quest to find formulas for solving polynomial equations. While general formulas exist for polynomials of degree up to 4, Galois proved that no such formula exists for general polynomials of degree 5 or higher, resolving a problem that had puzzled mathematicians for centuries. 7.5.2 Field Extensions Definition A field is an extension of a field if is a subfield of , denoted as or . Types of Extensions Simple Extension: for some Algebraic Extension: Every element of is algebraic over (satisfies a polynomial with coefficients in ) Transcendental Extension: Contains at least one element that is not algebraic over Finite Extension: has finite dimension as a vector space over Degree of Extension The degree of a field extension , denoted , is the dimension of as a vector space over . Tower Law If are fields, then . 7.5.3 Splitting Fields Definition A splitting field of a polynomial is the smallest field extension of in which factors completely into linear factors. Example The splitting field of over is , since in . Theorem Every polynomial has a unique (up to isomorphism) splitting field. 7.5.4 Galois Groups Definition Let be a field extension. The Galois group is the group of all field automorphisms of that fix every element of : Properties forms a group under composition of functions If is the splitting field of a separable polynomial over , then Examples where is complex conjugation where 7.5.5 The Fundamental Theorem of Galois Theory Statement Let be a finite Galois extension with Galois group . There is a one-to-one correspondence between: The intermediate fields (where ) The subgroups of This correspondence is given by: For a subgroup , the corresponding field is For an intermediate field , the corresponding subgroup is Properties of the Correspondence if and only if (order-reversing) and is Galois if and only if is a normal subgroup of If is normal in , then 7.5.6 Solvability by Radicals Definition A polynomial is solvable by radicals if its roots can be expressed using field operations and taking nth roots. Theorem A polynomial is solvable by radicals if and only if the Galois group of its splitting field is a solvable group. Consequence The general polynomial of degree 5 or higher is not solvable by radicals because the symmetric group is not solvable for . 7.5.7 Applications of Galois Theory Classical Problems of Antiquity Galois theory provides elegant proofs for the impossibility of: Trisecting an arbitrary angle using straightedge and compass Doubling a cube using straightedge and compass Squaring a circle using straightedge and compass Constructing regular n-gons for certain values of n Insolvability of the Quintic Galois theory proves that there is no general formula using radicals for solving polynomial equations of degree 5 or higher. Field Theory Applications Characterizing finite fields Understanding field automorphisms Classifying field extensions 7.5.8 Concrete Examples Cyclotomic Extensions The cyclotomic field , where is a primitive nth root of unity, has Galois group isomorphic to , the multiplicative group of integers modulo . Quadratic Extensions For where is a square-free integer: The non-identity automorphism maps 7.5.9 Modern Developments Galois theory has evolved beyond its classical form to include: Infinite Galois theory Differential Galois theory Étale cohomology Connections to representation theory Applications in algebraic number theory 7.5.10 Exercises Determine the Galois group of over . Prove that is not solvable by radicals. Find all subfields of and determine the corresponding subgroups of the Galois group. Show that a regular 17-gon is constructible with straightedge and compass. Prove that if is prime, then is irreducible over . Determine the splitting field and Galois group of over . Prove that is transcendental using Galois-theoretic methods.