⭕Groups and Geometries Unit 10 – Introduction to Galois Theory

Key Concepts and Definitions

• Galois theory studies the relationship between field extensions and group theory
• A field is an algebraic structure with addition, subtraction, multiplication, and division operations that satisfy certain axioms (commutative, associative, distributive, identity, and inverse properties)
• Field extensions are created by adding elements to a base field, resulting in a larger field containing the original field as a subfield
  • For example, the complex numbers $\mathbb{C}$ are a field extension of the real numbers $\mathbb{R}$, which in turn is a field extension of the rational numbers $\mathbb{Q}$
• A Galois extension is a field extension $E/F$ where $E$ is the splitting field of a separable polynomial over $F$
• The Galois group of a field extension $E/F$, denoted $\text{Gal}(E/F)$, is the group of automorphisms of $E$ that fix $F$ pointwise
• A polynomial is separable if it has distinct roots in its splitting field
• The splitting field of a polynomial $f(x)$ over a field $F$ is the smallest field extension of $F$ containing all the roots of $f(x)$
• The degree of a field extension $E/F$, denoted $[E:F]$, is the dimension of $E$ as a vector space over $F$

Historical Context and Development

• Galois theory is named after Évariste Galois, a French mathematician who laid the foundations for the field in the early 19th century
• Galois' work was motivated by the problem of solving polynomial equations by radicals, which had been a long-standing challenge in mathematics
• The development of Galois theory was influenced by the works of other mathematicians, such as Lagrange, Abel, and Cauchy
• Galois introduced the concept of a group (now known as a Galois group) to study the symmetries of the roots of a polynomial equation
• Galois' ideas were not well-understood during his lifetime, and his work was only recognized and further developed decades after his untimely death at the age of 20
• The modern formulation of Galois theory, using the language of field extensions and automorphism groups, was developed by later mathematicians such as Dedekind and Artin
• Galois theory has since become a fundamental tool in various areas of mathematics, including number theory, algebraic geometry, and representation theory

Fundamental Theorems and Principles

• The Fundamental Theorem of Galois Theory establishes a one-to-one correspondence between the intermediate fields of a Galois extension and the subgroups of its Galois group
  • Specifically, if $E/F$ is a Galois extension with Galois group $G$, then there is a bijection between the set of intermediate fields $K$ (satisfying $F \subseteq K \subseteq E$) and the set of subgroups $H$ of $G$
• The correspondence is given by the fixed field map, which associates each subgroup $H$ of $G$ with the fixed field $E^H = \{x \in E : \sigma(x) = x \text{ for all } \sigma \in H\}$
• The Galois correspondence preserves inclusions, meaning that if $H_1 \subseteq H_2$ are subgroups of $G$, then $E^{H_2} \subseteq E^{H_1}$
• The Fundamental Theorem also implies that the Galois group of a Galois extension is always a finite group
• The Primitive Element Theorem states that if $E/F$ is a finite separable extension, then there exists an element $\alpha \in E$ (called a primitive element) such that $E = F(\alpha)$
• The Galois group of a polynomial $f(x)$ over a field $F$ is isomorphic to the Galois group of its splitting field extension over $F$
• The Abel-Ruffini Theorem, a consequence of Galois theory, states that there is no general algebraic solution (using radicals) for polynomial equations of degree 5 or higher

Field Extensions and Their Properties

• A field extension $E/F$ is a field $E$ containing a subfield $F$
• The elements of $E$ that are roots of polynomials with coefficients in $F$ are called algebraic over $F$; otherwise, they are called transcendental
• An algebraic extension is a field extension in which every element is algebraic over the base field
• A finite extension is a field extension with finite degree $[E:F]$
  • The degree of a finite extension is equal to the dimension of $E$ as a vector space over $F$
• A simple extension is a field extension generated by a single element, i.e., $E = F(\alpha)$ for some $\alpha \in E$
• A normal extension is a field extension $E/F$ where every irreducible polynomial in $F[x]$ that has a root in $E$ splits completely in $E[x]$
• A separable extension is a field extension in which every element is separable over the base field, meaning that its minimal polynomial has distinct roots
• A Galois extension is a field extension that is both normal and separable
• The compositum of two field extensions $E_1/F$ and $E_2/F$ is the smallest field containing both $E_1$ and $E_2$, denoted $E_1E_2$

Galois Groups and Their Structure

• The Galois group of a field extension $E/F$, denoted $\text{Gal}(E/F)$, is the group of automorphisms of $E$ that fix $F$ pointwise
  • An automorphism is a bijective homomorphism from a field to itself
• The Galois group of a Galois extension is always a finite group
• The order of the Galois group is equal to the degree of the extension, i.e., $|\text{Gal}(E/F)| = [E:F]$
• The Galois group acts transitively on the roots of an irreducible polynomial in the base field
  • Transitivity means that for any two roots $\alpha$ and $\beta$, there exists an automorphism $\sigma \in \text{Gal}(E/F)$ such that $\sigma(\alpha) = \beta$
• The Galois group of a polynomial is a subgroup of the symmetric group on its roots
  • The symmetric group $S_n$ is the group of all permutations of $n$ elements
• The structure of the Galois group provides information about the solvability of the corresponding polynomial equation by radicals
  • A polynomial is solvable by radicals if its Galois group is a solvable group (a group with a subnormal series whose factors are all abelian)
• The Fundamental Theorem of Galois Theory relates the subgroup structure of the Galois group to the intermediate fields of the extension

Applications in Polynomial Equations

• Galois theory provides a framework for determining the solvability of polynomial equations by radicals
• A polynomial equation $f(x) = 0$ is solvable by radicals over a field $F$ if and only if the Galois group of $f(x)$ over $F$ is a solvable group
• The Abel-Ruffini Theorem, a consequence of Galois theory, states that there is no general algebraic solution (using radicals) for polynomial equations of degree 5 or higher
• Galois theory can be used to prove the impossibility of certain geometric constructions with compass and straightedge alone
  • For example, the impossibility of trisecting an arbitrary angle or doubling the cube
• The Galois group of a polynomial determines the symmetries and relationships among its roots
  • For instance, if the Galois group is cyclic, then the roots can be expressed in terms of a single primitive element
• Galois theory provides a method for constructing regular polygons with compass and straightedge
  • A regular $n$-gon is constructible if and only if $n$ is a product of distinct Fermat primes (primes of the form $2^{2^k} + 1$) and a power of 2
• The study of Galois groups of polynomials over $\mathbb{Q}$ has applications in number theory, such as the study of Diophantine equations and the classification of finite extensions of $\mathbb{Q}$

Connections to Other Mathematical Areas

• Galois theory has deep connections to various branches of mathematics, including:
  • Number theory: Galois groups of polynomials over $\mathbb{Q}$ provide information about the arithmetic properties of their roots and related number fields
  • Algebraic geometry: Galois theory is used to study the symmetries and properties of algebraic varieties and their function fields
  • Representation theory: Galois groups can be studied through their linear representations, which relate to the representation theory of finite groups
• The Langlands program, a vast web of conjectures connecting number theory, representation theory, and algebraic geometry, heavily relies on Galois-theoretic concepts
• Galois theory has been generalized to infinite extensions, leading to the development of infinite Galois theory and its applications in algebraic geometry and number theory
• The Galois correspondence between intermediate fields and subgroups has analogues in other areas of mathematics, such as the Galois connection in order theory and the Galois correspondence in algebraic topology
• Galois theory has inspired the development of similar theories in other algebraic structures, such as Galois theories for rings, modules, and differential equations
• The ideas of Galois theory have been applied to the study of covering spaces in topology, where the fundamental group plays a role analogous to the Galois group

Problem-Solving Techniques and Examples

• When solving problems in Galois theory, it is essential to identify the base field, the polynomial or field extension under consideration, and the relevant Galois groups
• To determine the Galois group of a polynomial, one can:
  • Factor the polynomial over the base field to find its roots and their multiplicities
  • Determine the splitting field of the polynomial by adjoining the roots to the base field
  • Find the automorphisms of the splitting field that fix the base field pointwise
• Example: Consider the polynomial $f(x) = x^3 - 2$ over $\mathbb{Q}$. Its splitting field is $\mathbb{Q}(\sqrt[3]{2}, \omega)$, where $\omega$ is a primitive third root of unity. The Galois group is isomorphic to $S_3$, the symmetric group on three letters
• To find the intermediate fields of a Galois extension, one can use the Fundamental Theorem of Galois Theory and look for subgroups of the Galois group
  • Each subgroup corresponds to a unique intermediate field, which can be found by computing the fixed field of the subgroup
• Example: For the extension $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}$, the Galois group is isomorphic to the dihedral group $D_4$. The intermediate fields correspond to the subgroups of $D_4$, which include $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(i\sqrt[4]{2})$
• When working with polynomials, it is often helpful to use the properties of the Galois group to simplify expressions or prove identities involving the roots
  • For example, if the Galois group is cyclic, the roots can be expressed in terms of a single primitive element, which can simplify computations
• Example: The polynomial $x^4 - 5x^2 + 6$ has Galois group isomorphic to the dihedral group $D_4$. Using the symmetries of the roots under the action of $D_4$, one can show that the sum of the squares of the roots is equal to 10
• In geometric construction problems, Galois theory can be used to determine the constructibility of certain objects or to prove the impossibility of certain constructions
  • The key idea is to relate the geometric problem to a polynomial equation and study its Galois group
• Example: To prove that an angle of $20^\circ$ is not constructible with compass and straightedge, one can consider the polynomial $x^3 - 3x - 1$, which has $\cos(20^\circ)$ as a root. The Galois group of this polynomial is $S_3$, which is not solvable, implying that the angle is not constructible