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.
Finite Galois extensions and
Definition
A finite extension is Galois when it is normal (A normal algebraic extension is one in which every minimal polynomial with a root in the extension splits there) and separable (Separable algebraic elements and separable extensions). Its Galois group is
with the group operation supplied by Relative field automorphisms and . The notation is reserved here for an extension already known to be finite Galois; for an arbitrary extension the notation remains .
Depends on
Used by
- A finite Galois extension is Galois over every intermediate field Corollary
- The Galois closure of a finite separable extension Definition
- The Galois group of a separable polynomial Definition
- Equivalent characterizations of a finite Galois extension Theorem
- The fundamental theorem of finite Galois theory Theorem
Dependency tree · two levels
8 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, Definition 3.9 (standard reference, not scraped)
- K. Conrad, The Galois Correspondence, Definition 4.4 (standard reference, not scraped)