15.1 Galois connections and Galois theory

Galois connections link two partially ordered sets through monotone functions, establishing a correspondence between their elements. This powerful concept finds applications in various mathematical fields, from algebra to topology, allowing for the transfer of properties between different structures.

In Galois theory, these connections relate field extensions to subgroups of the Galois group. This relationship forms the foundation for the Fundamental Theorem of Galois Theory, providing a deep insight into the structure of field extensions and their symmetries.

Galois Connections and Galois Theory

Definition of Galois connections

• Pair of monotone functions between two partially ordered sets establishes correspondence between their elements
  • Given partially ordered sets $(A, \leq)$ and $(B, \leq)$, Galois connection consists of two monotone functions $f: A \to B$ and $g: B \to A$
  • For all $a \in A$ and $b \in B$, $f(a) \leq b \iff a \leq g(b)$ holds
• Relates elements of one partially ordered set to elements of another
  • Allows transfer of properties and structures between the two sets (lattices, algebras)
  • Can establish isomorphisms between certain subsets of the partially ordered sets (closure operators, fixed points)

Applications in Galois theory

• Relates field extensions and subgroups of the Galois group
  • Given field extension $L/K$, Galois group $\text{Gal}(L/K)$ is group of automorphisms of $L$ that fix $K$ (symmetries, permutations)
• Galois connection between partially ordered sets of subfields of $L$ containing $K$ and subgroups of $\text{Gal}(L/K)$
  • Monotone functions:
    • $f: \text{Subfields}(L/K) \to \text{Subgroups}(\text{Gal}(L/K))$, $f(E) = \{\sigma \in \text{Gal}(L/K) : \sigma(x) = x, \forall x \in E\}$ (fixed field)
    • $g: \text{Subgroups}(\text{Gal}(L/K)) \to \text{Subfields}(L/K)$, $g(H) = \{x \in L : \sigma(x) = x, \forall \sigma \in H\}$ (invariant subfield)
• Establishes one-to-one correspondence between subfields of $L$ containing $K$ and subgroups of $\text{Gal}(L/K)$ (intermediate fields, normal subgroups)

Fundamental theorem proof

• Using language of category theory:
  • For Galois extension $L/K$ with Galois group $G$, equivalence of categories between category of intermediate fields of $L/K$ and category of $G$-sets
• Proof relies on Galois connection between subfields of $L$ containing $K$ and subgroups of $G$
  • Functor from intermediate fields to $G$-sets sends subfield $E$ to set $G/H$, where $H = \{\sigma \in G : \sigma(x) = x, \forall x \in E\}$ (cosets, orbits)
  • Functor from $G$-sets to intermediate fields sends $G$-set $X$ to field $L^{G_x}$, where $G_x = \{\sigma \in G : \sigma(x) = x\}$ is stabilizer of element $x \in X$ (fixed points)
• Natural isomorphisms between the two functors constructed using Galois connection prove equivalence of categories (adjunctions, unit and counit)

Examples in algebra and geometry

• Galois connections appear in various algebraic and geometric contexts:
  • Between subgroups of a group and subsets of group closed under conjugation (normal subgroups, conjugacy classes)
  • Between submodules of a module and subsets of endomorphism ring (annihilators, centralizers)
  • Between subspaces of vector space and subspaces of its dual space (orthogonal complements, annihilators)
  • Between closed subsets of topological space and ideals of its ring of continuous functions (zero sets, vanishing ideals)
• Example in group theory:
  • In group $G$, Galois connection between subgroups of $G$ and subsets of $G$ closed under conjugation
  • Monotone functions:
    1. $f: \text{Subgroups}(G) \to \text{ConjugationClosedSubsets}(G)$, $f(H) = \bigcup_{g \in G} gHg^{-1}$ (normal closure)
    2. $g: \text{ConjugationClosedSubsets}(G) \to \text{Subgroups}(G)$, $g(S)$ is subgroup generated by $S$ (normal core)