Decomposition, inertia, and Frobenius elements

Let L/K be a finite Galois extension of number fields with group G, let $\mathfrak p$ be a nonzero prime of $\mathcal O_K$, and choose $\mathfrak P$ above it. The decomposition group consists of automorphisms stabilizing that prime ideal. Such automorphisms act on the residue extension $k(\mathfrak P)/k(\mathfrak p)$. The inertia group is the kernel of this residue action, not the subgroup fixing every element of L. The residue fields are finite, so their Galois group is cyclic.

The reduction map is surjective in this finite Galois setting. Its kernel has order e, the ramification index, and the residue group has order f, so the decomposition group has order ef. The full Galois group acts transitively on primes above the base prime; their number g satisfies |G|=efg. These uniform equalities use the Galois hypothesis. In a general non-Galois number-field extension one keeps separate e and f values and uses their sum-of-products identity instead.

Let q be the cardinality of $k(\mathfrak p)$. Arithmetic residue Frobenius is the automorphism $x\mapsto x^q$. If the prime is unramified, inertia is trivial, so this automorphism has a unique lift to the decomposition group, the arithmetic Frobenius element at the chosen prime. It satisfies $\sigma(a)\equiv a^q\pmod{\mathfrak P}$ for integer elements a. Its order is the residue degree f. The convention is part of the definition rather than a universally unspoken choice.

Changing the chosen prime above the same base prime conjugates the element by the corresponding Galois automorphism. Thus a base unramified prime determines a conjugacy class in a nonabelian Galois group, not a distinguished individual element. In an abelian group conjugacy is trivial, so the dependence disappears. Geometric Frobenius is the inverse of the arithmetic element. Reversing convention reverses powers in examples and must be reflected consistently in any representation or reciprocity formula that uses it.

At a ramified prime, residue Frobenius remains a well-defined element of the quotient decomposition group modulo inertia. Its preimages in the decomposition group differ by inertia elements, so there is generally no distinguished lift without additional structure or a separate choice. Calling one arbitrary lift the unique Frobenius would erase precisely the ramification information represented by the kernel. A trivial residue extension can therefore coexist with a nontrivial inertia group and a nontrivial local field extension.

Cyclotomic extensions give explicit unramified examples: at rational primes not dividing the conductor, arithmetic Frobenius sends a root of unity to its p-th power over Q. Its order is a multiplicative order in the appropriate unit group, not the additive period of a residue class. The associated residue degree and number of primes follow from the Galois size identity. Such concrete calculations make good tests for sign and inverse conventions, but they do not justify assigning a canonical global element to every ramified prime.