DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

Conjugate algebraic elements over a field

Definition

Let $α$ and $β$ be elements of field extensions of $F$ , both algebraic over $F$ . They are conjugate over $F$ when they have the same minimal polynomial over $F$ (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element), or equivalently when $β$ is a root of the minimal polynomial of $α$ . The relation is relative to the chosen base field. Relative embeddings and automorphisms are those of $F$ -homomorphisms and $F$ -embeddings of field extensions.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 19 results over 6 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)