🏃🏽‍♀️Galois Theory Unit 8 – The Fundamental Theorem of Galois Theory

Key Concepts and Definitions

• Galois group the group of automorphisms of a field extension that fixes the base field
• Field extension a larger field that contains a given base field
• Splitting field the smallest field extension of a base field over which a given polynomial splits into linear factors
• Normal extension a field extension that is the splitting field of a family of polynomials
• Separable polynomial a polynomial whose roots are distinct in an algebraic closure
  • Separable extension a field extension generated by a separable polynomial
• Fixed field the subfield of elements fixed by a group of automorphisms
• Correspondence a bijective map between intermediate fields and subgroups of the Galois group

Historical Context and Development

• Évariste Galois (1811-1832) developed the theory in the early 19th century
  • Galois aimed to solve the long-standing problem of finding a general solution to polynomial equations of degree 5 or higher
• Building upon the works of Abel, Ruffini, and Lagrange on permutations and equations
• Galois introduced the concept of a group (now called Galois group) to study field extensions
  • He established a connection between the structure of these groups and the solvability of equations
• Tragically, Galois died at the age of 20 in a duel, leaving behind unpublished manuscripts
• Later mathematicians, such as Liouville and Jordan, recognized the importance of Galois' work and further developed the theory

Statement of the Fundamental Theorem

• The Fundamental Theorem of Galois Theory establishes a correspondence between the intermediate fields of a Galois extension and the subgroups of its Galois group
• Specifically, for a Galois extension $L/K$ with Galois group $G$, the theorem states:
  1. There is a one-to-one correspondence between the intermediate fields of $L/K$ and the subgroups of $G$
  2. The correspondence reverses inclusions, i.e., if $H_1 \subseteq H_2$ are subgroups of $G$, then $L^{H_2} \subseteq L^{H_1}$, where $L^H$ denotes the fixed field of $H$
  3. $[L:L^H] = |H|$ and $[L^H:K] = [G:H]$, where $[L:K]$ denotes the degree of the extension and $|H|$ is the order of the subgroup
• The theorem provides a powerful tool for understanding the structure of field extensions and their associated Galois groups

Main Components and Their Relationships

• Galois extension a field extension $L/K$ such that $L$ is the splitting field of a separable polynomial over $K$
  • Equivalently, $L/K$ is normal and separable
• Galois group $\text{Gal}(L/K)$ the group of $K$-automorphisms of $L$, i.e., automorphisms of $L$ that fix $K$
• Intermediate fields fields $F$ such that $K \subseteq F \subseteq L$
  • The Galois correspondence maps each intermediate field $F$ to its Galois group $\text{Gal}(L/F)$, a subgroup of $\text{Gal}(L/K)$
• Fixed fields for a subgroup $H \leq \text{Gal}(L/K)$, the fixed field $L^H = {x \in L : \sigma(x) = x \text{ for all } \sigma \in H}$
  • The Galois correspondence maps each subgroup $H$ to its fixed field $L^H$, an intermediate field of $L/K$

Proof Outline and Key Steps

1. Show that the fixed field of a subgroup is an intermediate field
   • Let $H \leq \text{Gal}(L/K)$ and show that $K \subseteq L^H \subseteq L$
2. Show that the Galois group of an intermediate field is a subgroup
   • Let $F$ be an intermediate field of $L/K$ and show that $\text{Gal}(L/F) \leq \text{Gal}(L/K)$
3. Prove that the correspondence is bijective
   • Show that $H = \text{Gal}(L/L^H)$ for any subgroup $H \leq \text{Gal}(L/K)$
   • Show that $F = L^{\text{Gal}(L/F)}$ for any intermediate field $F$ of $L/K$
4. Establish the properties of the correspondence
   • Prove that the correspondence reverses inclusions
   • Prove the formulas relating degrees and orders: $[L:L^H] = |H|$ and $[L^H:K] = [G:H]$

Applications and Examples

• Solving polynomial equations determining the solvability of a polynomial equation by radicals based on the structure of its Galois group
  • Example: the general quintic equation $ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0$ is not solvable by radicals because its Galois group is $S_5$, which is not solvable
• Constructibility problems using Galois theory to determine which geometric constructions are possible with compass and straightedge
  • Example: the angle $\pi/7$ is not constructible because the Galois group of the splitting field of $x^7 - 1$ over $\mathbb{Q}$ is not a 2-group
• Algebraic number theory studying the Galois groups of extensions of $\mathbb{Q}$ and their connections to number-theoretic properties
  • Example: the Galois group of the splitting field of $x^3 - 2$ over $\mathbb{Q}$ is $S_3$, which provides information about the arithmetic of $\sqrt[3]{2}$
• Inverse Galois problem the question of which groups can occur as Galois groups of extensions of a given field (usually $\mathbb{Q}$)
  • Example: the cyclic group $C_n$ is realizable as a Galois group over $\mathbb{Q}$ for any $n$, as seen in the splitting field of $x^n - 1$

Common Misconceptions and Pitfalls

• Not every field extension is Galois the Fundamental Theorem only applies to Galois extensions, which are normal and separable
• The Galois group is not always abelian even for simple extensions like $\mathbb{Q}(\sqrt[3]{2})$, the Galois group is non-abelian ($S_3$)
• The Galois correspondence is not always trivial for a Galois extension $L/K$, there may be proper intermediate fields between $K$ and $L$
  • Example: $\mathbb{Q}(\sqrt{2}, \sqrt{3})$ has three proper intermediate fields: $\mathbb{Q}(\sqrt{2})$, $\mathbb{Q}(\sqrt{3})$, and $\mathbb{Q}(\sqrt{6})$
• Solvability by radicals is not equivalent to solvability of the Galois group a polynomial can have a solvable Galois group but not be solvable by radicals (e.g., $x^5 - x + 1$)
• The Galois group depends on the base field the same polynomial can have different Galois groups over different base fields
  • Example: $x^2 - 2$ has Galois group $C_2$ over $\mathbb{Q}$ but trivial Galois group over $\mathbb{R}$

Further Implications and Extensions

• Inverse Galois problem determining which groups can be realized as Galois groups over a given field
  • The problem is still open for many classes of groups and fields
• Galois representations studying the action of Galois groups on vector spaces and relating them to geometric and arithmetic properties
• Galois cohomology using cohomological methods to study Galois modules and their connections to algebraic number theory and arithmetic geometry
• Differential Galois theory extending the ideas of classical Galois theory to the context of differential equations
  • Analogous to the classical case, differential Galois theory relates the solvability of differential equations to the structure of differential Galois groups
• Grothendieck's Galois theory a vast generalization of Galois theory to the context of schemes and étale fundamental groups
  • This approach has deep connections to algebraic geometry, number theory, and topology