CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.

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Galois prime decomposition efg

Statement

Let $L / K$ be a finite Galois extension of number fields and let $p$ be a nonzero prime of $O_{K}$ . Every prime $P$ of $O_{L}$ above $p$ has the same ramification index $e$ and residue degree $f$ . If there are $g$ such primes, then $e f g = [L : K]$ .

Facts & Assumptions

Given: The data and hypotheses of the statement.

[F1]

Galois action on primes above a prime is transitive: Let L/K be a finite Galois extension of number fields and p a nonzero prime of $O_{K}$ . Then $G = Gal (L / K)$ acts transitively on the primes P above p.

[F2]

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

Proof

1.1

If $σ P = P^{'}$ , applying $σ$ to the unique factorization of $p O_{L}$ preserves the exponent of P. The induced map $O_{L} / P \to O_{L} / P^{'}$ fixes $O_{K} / p$ and preserves the residue degree. Transitivity therefore makes both e and f constant.

2.1

The fundamental identity becomes $\sum_{P ∣ p} e f = g e f = [L : K]$ . All three integers are positive; degree one gives e=f=g=1.

F2step 1.1∎

Depends on

Used by

Complete splitting and trivial frobenius Corollary
Decomposition inertia in a quadratic field Example
Frobenius in a small cyclotomic field Example
Decomposition group and completion Theorem

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

§9.2, Theorem 9.2.2 (standard reference, not scraped)