Theorem

Fundamental Theorem of Galois Theory

Algebra

Establishes an order-reversing bijection between the intermediate fields of a finite Galois field extension and the subgroups of its Galois group. Building on the work of Evariste Galois, it turns questions about field extensions and polynomial solvability into questions about finite group structure.

Facts

Statement

For a finite Galois field extension E over F, there is a one-to-one, inclusion-reversing correspondence between the intermediate fields of the extension and the subgroups of its Galois group. 1

Classification

Statement Form

Existence Theorem 1

Connections

Associated With

Abel-Ruffini Theorem

Has Statement Form

Existence Theorem

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

In Branch

Algebra

Proved By

Evariste Galois

Sources

1. Fundamental Theorem of Galois Theory (Wikipedia)

Wikimedia FoundationLead section, first sentence

Quote, Lead section, first sentence

the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups.

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Fundamental Theorem of Galois Theory

Associated With

Has Statement Form

In Branch

Proved By