Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06
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.

Trace and norm of an algebraic integer

Statement

If α∈OK, then Tr⁡K/Q(α) and NK/Q(α) belong to Z.

Facts & Assumptions

Given: α∈OK.

[F1]

Trace and norm are the trace and determinant of multiplication (The norm NK/F and trace Tr⁡K/F of a finite field extension).

[F2]

The monic minimal polynomial of an algebraic integer has coefficients in Z (Minimal-polynomial criterion for algebraic integers).

Proof

technique · direct
1.1

Let f(X)=Xr+cr−1Xr−1+⋯+c0 be the minimal polynomial of α over Q. Fact [F2] gives cj∈Z.

F2given
2.1

Put h=[K:Q(α)]. Regarding K as a Q(α)-vector space, multiplication by α acts as the scalar α on each of h basis directions. Hence its characteristic polynomial over Q is f(X)h. By [F1], Tr⁡K/Q(α)=−hcr−1,NK/Q(α)=(−1)rhc0h, which are integers.

F1step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

17 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