TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The fundamental identity for primes

Statement

For finite $L / K$ and nonzero $p \subseteq O_{K}$ , $\sum_{P ∣ p} e (P / p) f (P / p) = [L : K] .$

Proof

Given: the finite factorisation of $p O_{L}$ .

1.1

Localise $O_{L}$ at $p$ . It is a finite torsion-free module over the DVR $(O_{K})_{p}$ , hence free of rank $[L : K]$ . Filtering its reduction modulo $p$ by the finite prime-power factors, the $e (P / p)$ layers at $P$ are copies of its residue field.

givenalgebra

2.1

Taking dimensions over $O_{K} / p$ gives the left side, while the free rank in step 1.1 gives $[L : K]$ for the same quotient.

step 1.1algebra∎

Depends on

Used by

Splitting and ramification terminology Definition
Prime factorisation in quadratic fields Example
Ramification is detected by the number-field discriminant Theorem

Dependency tree · two levels

8 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, Theorem 3.34 (standard reference, not scraped)