Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13
How 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.

Any two splitting fields of a polynomial are isomorphic over the base field

Statement

If E/F and E′/F are splitting fields of the same nonzero polynomial f∈F[x], then there is a field isomorphism E→E′ that fixes F pointwise.

Facts & Assumptions

Given: Two splitting fields E/F and E′/F of the same nonzero polynomial.

[F1]

A base-field isomorphism extends to an isomorphism between splitting fields of the corresponding transported polynomials (A base-field isomorphism extends to an isomorphism between splitting fields of corresponding polynomials).

Proof

technique · direct
1.1

Apply [F1] to the identity isomorphism of F. It transports f to itself and therefore extends to an isomorphism E→E′ fixing F.

F1
2.1

This includes nonzero constants, whose splitting fields have empty root sets and both equal F.

F1step 1.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 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