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.
-homomorphisms and -embeddings of field extensions
Definition
Let and be field extensions (Field extensions, generated subrings , generated subfields , and simple extensions). An -homomorphism is a field homomorphism (Field homomorphism and embedding) satisfying for every . Because field homomorphisms are injective, it is also called an -embedding. A bijective -homomorphism is an -isomorphism, and an -isomorphism is an -automorphism of .
Depends on
Used by
- Conjugate algebraic elements over a field Definition
- The separable degree [K:F]ₛ as a count of embeddings into an algebraic closure Definition
- Restriction partitions embeddings in a finite tower into extension fibres Lemma
- Assuming Choice, a base-field embedding extends across every algebraic extension Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 16 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- P. L. Clark, Field Theory, Chapters 3 to 5 (standard reference, not scraped)
- J. S. Milne, Fields and Galois Theory, Chapters 2, 3, and 5 (standard reference, not scraped)