Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-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.

Complete splitting and trivial frobenius

Statement

An unramified nonzero prime p in a finite Galois extension L/K splits completely if and only if its arithmetic Frobenius conjugacy class is the identity class.

Facts & Assumptions

Given: The data and hypotheses of the statement.

[F1]

Frobenius order is residue degree: For finite Galois L/K and nonzero P∣p, the arithmetic Frobenius coset has order f(P/p) in D/I. If P is unramified, Frob⁡P has the same order in D.

[F2]

Galois prime decomposition efg: For a finite Galois extension L/K and nonzero prime p, every P above p has the same ramification index e and residue degree f. If there are g such primes, then efg=[L:K].

Proof

1.1

If p splits completely, all residue degrees are one. The Frobenius order is then one, so every Frobenius element is identity.

F1
2.1

Conversely identity Frobenius has order one, so f=1. Unramifiedness gives e=1, and efg=[L:K] now gives g=[L:K]. Thus the factorization has degree-many distinct primes of residue degree one, namely complete splitting.

F1F2∎

Depends on

Used by

Nothing in the library uses this result yet.

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