The 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 .
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 and subgroups of its Galois group
For a 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 K, [E:K]=∣Gal(E/K)∣ and [K:F]=[Gal(E/F):Gal(E/K)]
Asserts K becomes a of F if and only if Gal(E/K) forms a 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 of H, containing all elements of E fixed by every 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 (Q(2,3) corresponds to a smaller subgroup than Q(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(Q(2,3)/Q), Gal(Q(2,3)/Q(2)) fixes 2 but may permute 3
For subgroup H of Gal(E/F), corresponding intermediate field Fix(H) represents the fixed field of H
Example: In Gal(Q(2,3)/Q), the subgroup fixing 2 corresponds to the field Q(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=Q(2,3), intersection of Q(2) and Q(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: Q(2)/Q is Galois as it's the splitting field of x2−2
Identify all intermediate fields of the extension and determine their degrees over the base field
Example: For Q(2,3)/Q, intermediate fields include Q(2), Q(3), and Q(6)
Construct the lattice of intermediate fields, considering their inclusions and intersections
Example: In Q(2,3)/Q, Q⊂Q(2)⊂Q(2,3)
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 Q(2,3)/Q, ∣Gal(E/F)∣=[Q(2,3):Q]=4
Analyze the subgroup structure to identify the isomorphism class of the Galois group, often using knowledge of finite group theory
Example: Gal(Q(2,3)/Q) ≅V4 (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 by analyzing its Galois group
Example: Quintic polynomials with non-solvable Galois groups (S5 or A5) prove unsolvable by radicals
Apply the correspondence to find all intermediate fields of a given Galois extension
Example: For Q(32,ω)/Q, identify Q(32), Q(ω), and Q(−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 2+3 over Q(6) in Q(2,3)/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 Fpn over Fp
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