8.1 Statement and proof of the Fundamental Theorem

The Fundamental Theorem of Galois Theory is a game-changer in field theory. It links intermediate fields of a Galois extension to subgroups of its Galois group, creating a powerful tool for studying field extensions and their automorphisms.

This theorem simplifies finding intermediate fields and enables constructing extensions with specific properties. It bridges algebra and group theory, allowing us to solve problems in one realm by tackling them in the other.

Fundamental Theorem of Galois Theory

Statement and Implications

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• 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
• If L/K is a Galois extension with Galois group G, then there is a bijection between the set of intermediate fields F (with K ⊆ F ⊆ L) and the set of subgroups H of G
  • The bijection maps each intermediate field F to the subgroup Gal(L/F) of G, and each subgroup H of G to the fixed field L^H
• The Fundamental Theorem implies that the lattice of intermediate fields is isomorphic to the lattice of subgroups of the Galois group, with the order reversed
• The degree of an intermediate field extension F/K is equal to the index [G : Gal(L/F)] of the corresponding subgroup in the Galois group

Applications and Significance

• The Fundamental Theorem provides a powerful tool for studying the structure of field extensions and their automorphism groups
• It allows us to determine the degree of an extension by examining the corresponding subgroup of the Galois group (cyclotomic extensions, splitting fields of polynomials)
• The theorem simplifies the problem of finding all intermediate fields of a Galois extension to the problem of finding all subgroups of its Galois group
• It enables the construction of field extensions with desired properties by specifying the structure of the Galois group (constructing regular polygons, solving polynomial equations)
• The Fundamental Theorem has applications in various branches of mathematics, including algebraic number theory, algebraic geometry, and the theory of equations

Proving the Fundamental Theorem

Fixed Fields and Subgroups

• Step 1: Prove that if H is a subgroup of the Galois group G, then the fixed field L^H is an intermediate field of the extension L/K
• Step 2: Show that if F is an intermediate field of L/K, then the Galois group Gal(L/F) is a subgroup of G
• Step 3: Establish that the fixed field of Gal(L/F) is F itself, i.e., (L^Gal(L/F)) = F
• Step 4: Prove that the Galois group of the fixed field L^H is H itself, i.e., Gal(L/L^H) = H

Bijection and Degree Relation

• Step 5: Demonstrate that the correspondence between intermediate fields and subgroups of the Galois group is a bijection
  • Show that the maps F ↦ Gal(L/F) and H ↦ L^H are inverses of each other
  • Prove that the composition of these maps in either order yields the identity map (on intermediate fields or subgroups)
• Step 6: Prove that the degree of an intermediate field extension F/K is equal to the index [G : Gal(L/F)] of the corresponding subgroup in the Galois group
  • Use the fact that [L : K] = [L : F][F : K] and [L : K] = |G|
  • Apply Lagrange's theorem to relate the index of the subgroup to the degree of the extension

Significance of the Fundamental Theorem

Connecting Algebra and Group Theory

• The Fundamental Theorem provides a deep connection between the algebraic structure of field extensions and the group-theoretic structure of their automorphism groups
• It allows us to study the properties of field extensions by examining the corresponding subgroups of the Galois group, and vice versa
• The theorem establishes a dictionary between field-theoretic and group-theoretic concepts (intermediate fields ↔ subgroups, degree ↔ index, normal extensions ↔ normal subgroups)
• It enables the transfer of problems and techniques between the realms of field theory and group theory (solvability by radicals ↔ solvable groups)

Simplifying Problems and Computations

• The Fundamental Theorem simplifies the problem of finding all intermediate fields of a Galois extension to the problem of finding all subgroups of its Galois group
• It allows us to determine the degree of an intermediate field extension by computing the index of the corresponding subgroup in the Galois group
• The theorem provides a method for constructing field extensions with specific properties by controlling the structure of the Galois group (constructing regular polygons, solving polynomial equations)
• It enables the computation of Galois groups and the study of their structure using techniques from group theory (Sylow theorems, group actions)

Field Extension Degree vs Galois Group Order

Equality for Galois Extensions

• For a Galois extension L/K with Galois group G, the degree of the extension [L : K] is equal to the order of the Galois group |G|
• This relationship is a consequence of the Fundamental Theorem of Galois Theory
• It follows from the fact that the fixed field of the entire Galois group G is the base field K, i.e., L^G = K
• The equality [L : K] = |G| holds for any Galois extension, regardless of its specific structure or properties

Subgroup Index and Intermediate Field Degree

• If F is an intermediate field of L/K, then the degree of the extension F/K is equal to the index [G : Gal(L/F)] of the corresponding subgroup in the Galois group
• The degree of the extension L/F is equal to the order of the subgroup Gal(L/F)
• These relationships follow from the Fundamental Theorem and the multiplicativity of field extension degrees, i.e., [L : K] = [L : F][F : K]
• The relationship between subgroup index and intermediate field degree allows us to compute the degree of an extension by studying its Galois group (using Lagrange's theorem, group actions)