Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
A finite Galois extension has finitely many intermediate fields
Statement
A finite Galois extension has only finitely many intermediate fields.
Facts & Assumptions
Given: A finite Galois extension and its finite Galois group .
The assignments and are mutually inverse inclusion-reversing bijections (The fundamental theorem of finite Galois theory).
Proof
A finite group has a finite power set, and its subgroups form a subcollection of that power set; hence has finitely many subgroups.
By [L1], the intermediate fields are in bijection with those subgroups, so there are finitely many. When , both collections have one member, and the base and top endpoints coincide.
Remarks
The library proves more than this elsewhere: A finite separable extension has only finitely many intermediate fields drops normality and keeps the conclusion, by the Steinitz primitive-element route rather than by the correspondence. The corollary here is recorded because it is what the Galois correspondence gives immediately, not because the separable statement is unavailable.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. S. Milne, Fields and Galois Theory, v5.10, Theorem 3.17 (standard reference, not scraped)
- K. Conrad, The Galois Correspondence, Theorem 5.6 (standard reference, not scraped)