10.3 The Fundamental Theorem of Galois Theory

The Fundamental Theorem of Galois Theory is the cornerstone of Galois theory, linking field extensions to group theory. It sets up a one-to-one correspondence between intermediate fields of a Galois extension and subgroups of its Galois group.

This theorem is crucial for understanding the structure of field extensions and solving polynomial equations. It allows us to translate complex field theory problems into more manageable group theory questions, making it a powerful tool in abstract algebra.

The Fundamental Theorem of Galois Theory

Core Concepts and Correspondence

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• Establishes a bijective correspondence between intermediate fields of a field extension and subgroups of its Galois group
• For a finite Galois extension E/F, creates a one-to-one correspondence between intermediate fields K (F ⊆ K ⊆ E) and subgroups H of the Galois group Gal(E/F)
• Correspondence defined as K ↔ Gal(E/K) for intermediate fields K, and H ↔ Fix(H) for subgroups H of Gal(E/F)
• States that for any intermediate field K, $[E:K] = |Gal(E/K)|$ and $[K:F] = [Gal(E/F) : Gal(E/K)]$
• Asserts K becomes a normal extension of F if and only if Gal(E/K) forms a normal subgroup of Gal(E/F)
• Requires understanding of field theory, group theory, and concepts of Galois extensions and Galois groups

Mathematical Properties and Relationships

• Exhibits order-reversing nature with larger subfields corresponding to smaller subgroups, and vice versa
• For intermediate field K, corresponding subgroup Gal(E/K) contains all automorphisms in Gal(E/F) fixing every element of K
• For subgroup H of Gal(E/F), corresponding intermediate field Fix(H) represents the fixed field of H, containing all elements of E fixed by every automorphism in H
• Preserves inclusions with K₁ ⊆ K₂ implying Gal(E/K₂) ⊆ Gal(E/K₁), and H₁ ⊆ H₂ implying Fix(H₁) ⊇ Fix(H₂)
• Respects intersections and compositums of fields, as well as intersections and joins of subgroups
• Requires familiarity with lattice theory and its application to subfields and subgroups

Subfields vs Subgroups Correspondence

Field-Subgroup Relationships

• Demonstrates order-reversing property where larger subfields correspond to smaller subgroups ($\mathbb{Q}(\sqrt{2},\sqrt{3})$ corresponds to a smaller subgroup than $\mathbb{Q}(\sqrt{2})$)
• For intermediate field K, corresponding subgroup Gal(E/K) consists of automorphisms in Gal(E/F) fixing every element of K
  • Example: In Gal($\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}$), Gal($\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}(\sqrt{2})$) fixes $\sqrt{2}$ but may permute $\sqrt{3}$
• For subgroup H of Gal(E/F), corresponding intermediate field Fix(H) represents the fixed field of H
  • Example: In Gal($\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}$), the subgroup fixing $\sqrt{2}$ corresponds to the field $\mathbb{Q}(\sqrt{2})$

Lattice Structure and Properties

• Preserves inclusions with K₁ ⊆ K₂ implying Gal(E/K₂) ⊆ Gal(E/K₁), and H₁ ⊆ H₂ implying Fix(H₁) ⊇ Fix(H₂)
• Respects intersections and compositums of fields, as well as intersections and joins of subgroups
  • Example: For $E = \mathbb{Q}(\sqrt{2},\sqrt{3})$, intersection of $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ corresponds to join of their respective fixing subgroups
• Applies concepts from lattice theory to analyze relationships between subfields and subgroups
  • Example: Hasse diagram of subfields mirrors inverted Hasse diagram of subgroups

Galois Group Determination

Verification and Intermediate Field Analysis

• Verify the given extension qualifies as Galois (normal and separable)
  • Example: $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ is Galois as it's the splitting field of $x^2 - 2$
• Identify all intermediate fields of the extension and determine their degrees over the base field
  • Example: For $\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}$, intermediate fields include $\mathbb{Q}(\sqrt{2})$, $\mathbb{Q}(\sqrt{3})$, and $\mathbb{Q}(\sqrt{6})$
• Construct the lattice of intermediate fields, considering their inclusions and intersections
  • Example: In $\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}$, $\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset \mathbb{Q}(\sqrt{2},\sqrt{3})$

Galois Group Construction and Analysis

• Use the correspondence to construct the lattice of subgroups of the Galois group, matching the structure of the field lattice
• Determine the order of the Galois group using the degree of the extension: $|Gal(E/F)| = [E:F]$
  • Example: For $\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}$, $|Gal(E/F)| = [\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}] = 4$
• Analyze the subgroup structure to identify the isomorphism class of the Galois group, often using knowledge of finite group theory
  • Example: Gal($\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}$) $\cong V_4$ (Klein four-group) due to its structure and order

Applications of the Fundamental Theorem

Problem-Solving in Field Theory

• Determine whether a given polynomial becomes solvable by radicals by analyzing its Galois group
  • Example: Quintic polynomials with non-solvable Galois groups ($S_5$ or $A_5$) prove unsolvable by radicals
• Apply the correspondence to find all intermediate fields of a given Galois extension
  • Example: For $\mathbb{Q}(\sqrt[3]{2},\omega)/\mathbb{Q}$, identify $\mathbb{Q}(\sqrt[3]{2})$, $\mathbb{Q}(\omega)$, and $\mathbb{Q}(\sqrt{-3})$ as intermediate fields
• Determine the minimal polynomial of an element over an intermediate field using the Galois group and its subgroups
  • Example: Find minimal polynomial of $\sqrt{2} + \sqrt{3}$ over $\mathbb{Q}(\sqrt{6})$ in $\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}$

Theoretical Applications and Proofs

• Prove classical results in Galois theory, such as the impossibility of certain geometric constructions
  • Example: Demonstrate the impossibility of trisecting an arbitrary angle using straightedge and compass
• Apply the theorem to analyze the structure of finite fields and their automorphisms
  • Example: Determine the Galois group of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$
• Utilize the Galois correspondence to solve problems involving field-theoretic properties such as normality, separability, and simplicity of extensions
  • Example: Prove that a finite extension is normal if and only if it's the splitting field of a separable polynomial