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.

Inertia group of a prime

Definition

Let $L / K$ be a finite Galois extension of number fields, let $p$ be a nonzero prime of $O_{K}$ , and choose a prime $P$ of $O_{L}$ above $p$ . Set $κ (P) = O_{L} / P$ and $κ (p) = O_{K} / p$ . Each $σ \in D (P / p)$ induces a $κ (p)$ -automorphism $\overset{ˉ}{σ}$ of $κ (P)$ : the rule $\overset{ˉ}{a} \mapsto \overline{σ a}$ is independent of the representative because $σ P = P$ . The inertia group is $I (P / p) = \ker ⁣ (D (P / p) ⟶ Gal (κ (P) / κ (p))) .$ Equivalently, $σ \in I (P / p)$ exactly when $σ \in D (P / p)$ and $σ (a) - a \in P$ for every $a \in O_{L}$ . It is a normal subgroup of D.

Depends on

Used by

Good polynomial reduction kills inertia Lemma
Lifting residue frobenius by galois conjugates Lemma
Conjugacy of decomposition and inertia groups Theorem
Decomposition inertia exact sequence Theorem

Dependency tree · two levels

11 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

§9.3.2, Definition 9.3.6 and Proposition 9.3.8, p.106 (standard reference, not scraped)