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

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Unramified frobenius element exists uniquely

Statement

For finite Galois L/K and a nonzero prime $P ∣ p$ with $e (P / p) = 1$ , there is a unique ${Frob}_{P} \in D (P / p)$ satisfying ${Frob}_{P} (a) \equiv a^{N p} (m o d P) (a \in O_{L}) .$ It is the arithmetic Frobenius element, the unique lift of the arithmetic Frobenius coset.

Facts & Assumptions

Given: The data and hypotheses of the statement.

[F1]

Arithmetic frobenius coset: For finite Galois L/K and nonzero $P ∣ p$ , the arithmetic Frobenius coset is the unique element of $D (P / p) / I (P / p)$ corresponding under the residue isomorphism to $x \mapsto x^{N p}$ on $κ (P)$ , where $N p = ∣ κ (p) ∣$ . It is defined also when P is ramified. Its inverse is called geometric Frobenius. The quotient element is distinguished; a representative in D need not be unique.

[F2]

Orders of decomposition and inertia groups: For finite Galois L/K and nonzero $P ∣ p$ , writing e and f for its ramification index and residue degree, $∣ D (P / p) ∣ = e f, ∣ I (P / p) ∣ = e, ∣ D (P / p) / I (P / p) ∣ = f .$ The prime P is unramified over p if and only if its inertia group is trivial.

Proof

1.1

The assumption e=1 implies I is trivial. Consequently the quotient map $D \to D / I$ is an isomorphism of groups, including when D itself is trivial.

2.1

The coset defined by the arithmetic residue power map therefore has exactly one inverse image. Reduction of that image is the power map, which is precisely the displayed congruence on every residue representative, including zero. Conversely any element of D with those congruences has the same quotient image and hence equals it.

F1step 1.1∎

Depends on

Used by

Frobenius compatibility in finite towers Corollary
Frobenius cycle type and prime splitting Theorem
Frobenius elements above a prime are conjugate Theorem
Frobenius order is residue degree Theorem

Dependency tree · two levels

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Sources

Chapter 8, Frobenius element, pp.141–142 (standard reference, not scraped)