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Canonical static concept record

Fundamental theorem of Galois theory

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Summary

For a finite Galois extension, intermediate fields correspond inclusion-reversingly to subgroups of the Galois group; normal subextensions correspond to normal subgroups.

Record metadata

Canonical name: Fundamental theorem of Galois theory
Concept ID: fundamental_theorem_galois_theory
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebra
Domains: Algebra, Number theory
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

finite Galois extension

Data

fixed-field and automorphism-group assignments

Axioms / constraints

mutually inverse correspondence

Canonically induces

Degree-index relations and quotient Galois groups

Concept sources

Wikipedia — Fundamental theorem of Galois theory — Fundamental theorem of Galois theory · encyclopedic reference · source ID wp-fundamental-theorem-galois-theory
nLab — the fundamental theorem of Galois theory — the fundamental theorem of Galois theory · mathematical reference · source ID nlab-fundamental-theorem-galois-theory
Milne — Fields and Galois Theory — Fields and Galois Theory · course text · source ID milne-fields-galois-theory

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Galois extension Fundamental theorem of Galois theory

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_galois_extension_to_fundamental_theorem_galois_theory_theorem_implication
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: subgroups correspond to subfields

Authored explanation

Finite Galois extensions have an inclusion-reversing subgroup–intermediate-field correspondence.

How to interpret this relation type

Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.

Relation sources

Milne — Fields and Galois Theory — Fields and Galois Theory · course text · source ID milne-fields-galois-theory

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