🏃🏽‍♀️Galois Theory Unit 5 – Automorphisms and Fixed Fields

Key Concepts and Definitions

• Automorphism: a bijective homomorphism from a mathematical object to itself that preserves the object's structure
• Fixed field: the subfield of elements that remain unchanged under a given automorphism or group of automorphisms
• Galois correspondence: a fundamental theorem establishing a one-to-one correspondence between intermediate fields and subgroups of the Galois group
  • Relates the lattice of intermediate fields to the lattice of subgroups of the Galois group
• Field extension: a field $K$ containing a subfield $F$, denoted as $K/F$
• Galois extension: a field extension $K/F$ where $K$ is the splitting field of a separable polynomial over $F$
• Galois group: the group of all automorphisms of a Galois extension that fix the base field
• Intermediate field: a field $L$ satisfying $F \subseteq L \subseteq K$ for a field extension $K/F$
• Separable polynomial: a polynomial $f(x)$ over a field $F$ where all its roots in an algebraic closure have multiplicity 1

Automorphisms: Properties and Examples

• Automorphisms preserve field operations (addition, multiplication, and multiplicative inverses) and field axioms
• Examples of automorphisms include:
  • Identity automorphism: maps every element to itself
  • Complex conjugation: maps $a+bi$ to $a-bi$ in the complex numbers
• Automorphisms form a group under composition, with the identity automorphism as the identity element
  • Composition of automorphisms is associative: $(\sigma \circ \tau) \circ \rho = \sigma \circ (\tau \circ \rho)$
  • Every automorphism has an inverse automorphism: $\sigma \circ \sigma^{-1} = \sigma^{-1} \circ \sigma = id$
• Automorphisms are isomorphisms from a field to itself, preserving the field's algebraic structure
• The set of all automorphisms of a field extension $K/F$ that fix $F$ is denoted as $Aut(K/F)$

Group of Automorphisms

• The set of all automorphisms of a field extension $K/F$ that fix $F$, denoted as $Aut(K/F)$, forms a group under composition
  • The identity element is the identity automorphism
  • Inverse elements are the inverse automorphisms
  • Composition is associative
• $Aut(K/F)$ is a subgroup of the symmetric group on the set $K$
• The order of $Aut(K/F)$ is called the degree of the extension, denoted as $[K:F]$
• For a Galois extension $K/F$, the Galois group $Gal(K/F)$ is defined as $Aut(K/F)$
  • $Gal(K/F)$ is a finite group with order equal to $[K:F]$
• The Galois group acts on the roots of a polynomial $f(x)$ over $F$ by permutation

Fixed Fields: Basics and Significance

• The fixed field of an automorphism $\sigma$ of $K/F$ is the set $\{x \in K : \sigma(x) = x\}$
  • Elements in the fixed field remain unchanged under the automorphism
• The fixed field of a group of automorphisms $G$ is the intersection of the fixed fields of all automorphisms in $G$
  • $Fix(G) = \{x \in K : \sigma(x) = x \text{ for all } \sigma \in G\}$
• Fixed fields are always subfields of $K$ containing $F$
  • $F \subseteq Fix(G) \subseteq K$
• The fixed field of $Aut(K/F)$ is precisely the base field $F$
• For a Galois extension $K/F$, the fixed field of $Gal(K/F)$ is $F$
• Fixed fields play a crucial role in the Galois correspondence, relating subgroups of the Galois group to intermediate fields

Galois Correspondence

• The Galois correspondence is a one-to-one correspondence between the intermediate fields of a Galois extension $K/F$ and the subgroups of the Galois group $Gal(K/F)$
  • Each intermediate field $L$ corresponds to a unique subgroup $H$ of $Gal(K/F)$, and vice versa
• The correspondence is given by two maps:
  • $L \mapsto Gal(K/L)$, mapping an intermediate field to its corresponding subgroup
  • $H \mapsto Fix(H)$, mapping a subgroup to its fixed field
• Properties of the Galois correspondence:
  • $L_1 \subseteq L_2 \iff Gal(K/L_2) \subseteq Gal(K/L_1)$ (inclusion-reversing)
  • $[K:L] = |Gal(K/L)|$ (degree of extension equals order of corresponding subgroup)
  • $L/F$ is Galois $\iff$ $Gal(K/L)$ is a normal subgroup of $Gal(K/F)$
• The Galois correspondence provides a powerful tool for studying field extensions and their symmetries

Applications in Field Theory

• The Galois correspondence can be used to prove the fundamental theorem of Galois theory
  • Every finite separable extension of a field is a Galois extension
• Galois theory can be applied to solve classical problems in field theory, such as:
  • Proving the impossibility of certain geometric constructions with compass and straightedge (squaring the circle, trisecting an angle, doubling the cube)
  • Determining the solvability of polynomial equations by radicals
• The Galois group of a polynomial determines its solvability by radicals
  • A polynomial is solvable by radicals if and only if its Galois group is solvable
• Galois theory provides a framework for studying the symmetries and structure of field extensions
• Applications of Galois theory extend beyond field theory to areas such as algebraic geometry and number theory

Problem-Solving Techniques

• To find the Galois group of a polynomial $f(x)$ over a field $F$:
  1. Find the splitting field $K$ of $f(x)$ over $F$
  2. Determine the automorphisms of $K$ that fix $F$ by permuting the roots of $f(x)$
  3. The Galois group is the group of these automorphisms under composition
• To find the fixed field of a group of automorphisms $G$:
  1. Find the elements of $K$ that remain unchanged under all automorphisms in $G$
  2. Verify that the set of these elements forms a subfield of $K$ containing $F$
• To find the intermediate fields of a Galois extension $K/F$:
  1. Find all subgroups of the Galois group $Gal(K/F)$
  2. For each subgroup $H$, find its fixed field $Fix(H)$
  3. The intermediate fields are precisely the fixed fields of the subgroups
• Utilize the properties of the Galois correspondence to solve problems involving field extensions and their symmetries

Connections to Other Topics

• Galois theory is closely related to the theory of algebraic equations and the solvability of polynomials by radicals
  • The Abel-Ruffini theorem states that there is no general solution by radicals for polynomials of degree 5 or higher
• Galois groups can be used to study the symmetries of geometric objects, such as regular polygons and platonic solids
• The Galois correspondence has analogues in other areas of mathematics, such as:
  • The fundamental theorem of Galois theory for rings and modules
  • The Galois connection in order theory and lattice theory
• Galois theory has applications in cryptography, particularly in the design of secure cryptographic protocols
  • The security of certain cryptographic schemes relies on the difficulty of solving polynomial equations over finite fields
• The concepts and techniques of Galois theory have influenced the development of modern algebra and abstract algebra
  • Galois theory demonstrates the power of studying the symmetries and automorphisms of mathematical objects