Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24
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.

The F[x]-module VT of an endomorphism

Definition

Let V be a vector space over a field F and let T:V→V be an endomorphism. The polynomial module of T, denoted VT, is the additive group of V with the F[x]-action

p⋅v:=p(T)v

from Polynomial evaluation at an endomorphism: p(T)=∑kakTk. This is a unital left F[x]-module (Unital left and right modules over a ring; unqualified module means left module): distributivity follows from linearity, 1(T)=IV, and (pq)(T)=p(T)q(T) by expanding the finite polynomial sums. The subscript records the action; it does not change the underlying vectors or addition.

Depends on

Used by

Dependency tree · two levels

7 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