3.2 Galois groups and Galois correspondence

Galois groups and correspondence are key concepts in algebraic extensions. They link field theory to group theory, revealing deep connections between subfields and subgroups. This powerful tool helps us understand field structures and solve algebraic problems.

The Fundamental Theorem of Galois Theory establishes a one-to-one correspondence between intermediate fields and subgroups of the Galois group. This correspondence allows us to translate field properties into group properties and vice versa, providing insights into field extensions.

Galois groups of field extensions

Definition and properties

• Galois group of field extension K/F comprises all automorphisms of K fixing every element of F
• Order of Galois group for finite extension K/F limited to at most [K:F] (degree of extension)
• Normal extension K/F called Galois extension when |Gal(K/F)| equals [K:F]
• Galois group of separable polynomial f(x) over F isomorphic to subgroup of symmetric group Sn (n = degree of f(x))
• Splitting field K of separable polynomial f(x) over F has Gal(K/F) isomorphic to transitive subgroup of Sn
  • Example: Galois group of $x^3 - 2$ over Q isomorphic to S3
  • Example: Galois group of $x^4 + 1$ over Q isomorphic to D4 (dihedral group of order 8)

Fixed fields and structure

• Fixed field of subgroup H of Gal(K/F) contains all elements in K fixed by every automorphism in H
  • Example: Fixed field of subgroup ${id, (12)}$ in Gal(Q(√2,√3)/Q) is Q(√3)
• Galois group encodes information about field extension structure
  • Normality: K/F normal if and only if |Gal(K/F)| = [K:F]
  • Separability: K/F separable if and only if |Gal(K/F)| = [K:F]
• Galois group actions reveal algebraic relationships between field elements
  • Example: In Gal(Q(√2,√3)/Q), automorphism sending √2 to -√2 and fixing √3 shows algebraic independence of √2 and √3 over Q

Fundamental theorem of Galois theory

Correspondence between fields and subgroups

• Establishes one-to-one correspondence between intermediate fields of Galois extension K/F and subgroups of Gal(K/F)
• Bijection between lattice of intermediate fields E (F ⊆ E ⊆ K) and lattice of subgroups H of Gal(K/F)
• Correspondence given by E ↦ Gal(K/E) and H ↦ Fix(H) (fixed field of H)
• Inclusion reversal: E1 ⊆ E2 implies Gal(K/E1) ⊇ Gal(K/E2)
  • Example: For K = Q(√2,√3), E1 = Q(√2) and E2 = Q(√2,√3), Gal(K/E1) = {id, σ} and Gal(K/E2) = {id}, where σ(√3) = -√3

Field degrees and group orders

• For intermediate field E, [K:E] equals |Gal(K/E)|
• [E:F] equals [Gal(K/F) : Gal(K/E)] (index of subgroup)
• Gal(E/F) isomorphic to quotient group Gal(K/F)/Gal(K/E)
  • Example: For K = Q(ζ7) (ζ7 = e^(2πi/7)), E = Q(ζ7 + ζ7^-1), |Gal(K/Q)| = 6, |Gal(K/E)| = 2, so [E:Q] = 3

Normal extensions and subgroups

• Normal subgroups of Gal(K/F) correspond to normal extensions of F within K
  • Example: In Gal(Q(√2,√3)/Q) ≅ V4, normal subgroups correspond to Q(√2), Q(√3), and Q(√6)
• Allows construction of tower of fields with known Galois groups
  • Example: Q ⊂ Q(√2) ⊂ Q(√2,√3) with corresponding Galois groups C2 and V4

Galois correspondence

Algebraic properties and field structures

• Cyclic subgroups of Gal(K/F) correspond to simple algebraic extensions of F within K
  • Example: In Gal(Q(ζ5)/Q) ≅ C4, cyclic subgroup of order 2 corresponds to Q(√5)
• Abelian subgroups of Gal(K/F) correspond to abelian extensions of F (compositions of cyclic extensions)
  • Example: Q(ζn) over Q is abelian for any n, as Gal(Q(ζn)/Q) ≅ (Z/nZ)*
• Fixed field of commutator subgroup of Gal(K/F) maximal abelian subextension of K/F
  • Example: For K = splitting field of x^3 - 2 over Q, fixed field of commutator subgroup is Q(√-3)

Group theory and field theory connections

• Order of element in Gal(K/F) corresponds to degree of minimal polynomial of primitive element of corresponding fixed field over F
  • Example: In Gal(Q(ζ7)/Q), element of order 3 corresponds to cubic subfield Q(ζ7 + ζ7^-1)
• Galois correspondence translates group-theoretic problems into field-theoretic ones and vice versa
  • Example: Impossibility of trisecting an angle with compass and straightedge translated to non-existence of certain subgroups in Gal(Q(ζn)/Q)
• Normal subgroups of Gal(K/F) correspond to normal extensions of F within K, enabling analysis of solvability by radicals
  • Example: A5 as Galois group of general quintic polynomial explains its non-solvability by radicals

Computing Galois groups

Simple extensions and cyclotomic fields

• Galois group of simple algebraic extension K = F(α) isomorphic to subgroup of Sn (n = degree of minimal polynomial of α over F)
  • Example: Gal(Q(√2)/Q) ≅ C2, as minimal polynomial x^2 - 2 has degree 2
• Galois group of cyclotomic extension Q(ζn)/Q isomorphic to (Z/nZ)*
  • Example: Gal(Q(ζ5)/Q) ≅ (Z/5Z)* ≅ C4

Radical extensions and splitting fields

• Radical extension K = F(α^(1/n)) has Galois group as subgroup of semidirect product of cyclic groups
  • Example: Gal(Q(2^(1/3))/Q) ≅ S3, determined by splitting field of x^3 - 2 over Q
• Splitting field of separable polynomial f(x) over F has Galois group isomorphic to transitive subgroup of Sn
  • Example: Splitting field of x^4 + 1 over Q has Galois group D4 (dihedral group of order 8)

Composite extensions and techniques

• Galois group of composite extension K = F(α,β) analyzed using tensor product of fields
• Gal(K/F) ≅ Gal(F(α)/F) × Gal(F(β)/F) if F(α) and F(β) linearly disjoint over F
  • Example: Gal(Q(√2,√3)/Q) ≅ C2 × C2, as Q(√2) and Q(√3) linearly disjoint over Q
• Tower of extensions F ⊆ E ⊆ K analyzed using short exact sequence: 1 → Gal(K/E) → Gal(K/F) → Gal(E/F) → 1
  • Example: For Q ⊆ Q(√2) ⊆ Q(√2,√3), sequence is 1 → C2 → V4 → C2 → 1
• Techniques for computing Galois groups of specific polynomial extensions
  • Analyzing fixed fields (finding elements fixed by subgroups)
  • Using discriminants (determining transitivity and primitivity of Galois group)
  • Applying Kummer theory (for radical extensions of fields containing roots of unity)