DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Arithmetic frobenius coset

Definition

For finite Galois L/K and nonzero $P ∣ p$ , the arithmetic Frobenius coset is the unique element of $D (P / p) / I (P / p)$ corresponding under the residue isomorphism to $x \mapsto x^{N p}$ on $κ (P)$ , where $N p = ∣ κ (p) ∣$ . It is defined also when P is ramified. Its inverse is called geometric Frobenius. The quotient element is distinguished; a representative in D need not be unique.

Depends on

Decomposition inertia exact sequence

Used by

Ramified frobenius has no canonical lift Counterexample
Frobenius order is residue degree Theorem
Unramified frobenius element exists uniquely Theorem

Dependency tree · two levels

4 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

Chapter 8, Frobenius element, p.141 (standard reference, not scraped)