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.
Relative field automorphisms and
Definition
Let be a field extension. An -automorphism of is an -isomorphism . This is the relative-automorphism case of -homomorphisms and -embeddings of field extensions.
The set of all -automorphisms of is denoted
Composition is the proposed operation. That it makes this set a group, and the basic finite-extension bound on its order, are proved in is a group and ↗.
Depends on
Used by
- Fₚ(t)/Fₚ(tᵖ) is normal and inseparable with trivial automorphism group Counterexample
- ℚ(³√2)/ℚ is separable and nonnormal with trivial automorphism group Counterexample
- Finite Galois extensions and Gal(K/F) Definition
- The fixed field K^G of a group of field automorphisms Definition
- Aut(K/F) is a group and |Aut(K/F)|≤ [K:F]ₛ≤ [K:F] Theorem
Dependency tree · two levels
3 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, Chapter 3 (standard reference, not scraped)
- K. Conrad, The Galois Correspondence, Section 4 (standard reference, not scraped)