TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-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.

Ramification and residue degrees in towers

Statement

For $M / L / K$ and $Q ∣ P ∣ p$ , $e (Q / p) = e (Q / P) e (P / p), f (Q / p) = f (Q / P) f (P / p) .$

Proof

Given: the indicated tower and primes.

1.1givenalgebra

Substitute the prime factorisation of $p O_{L}$ into its extension to $O_{M}$ and compare the exponent of $Q$ .

2.1step 1.1algebra∎

The three residue fields form a finite tower, so the dimensions multiply. These are exactly the two asserted equalities.

Depends on

Used by

Frobenius compatibility in finite towers Corollary
Decomposition and inertia fixed fields Theorem
Decomposition and inertia in towers Theorem

Dependency tree · two levels

6 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

J. S. Milne, Algebraic Number Theory, Chapter 4, Exercise 4-2 (standard reference, not scraped)