Splitting of prime ideals in a Galois extension

In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers O_K factorise as products of prime ideals of O_L, provides one of the richest parts of algebraic number theory. The splitting of prime ideals in Galois extensions is sometimes attributed to David Hilbert by calling it Hilbert theory. There is a geometric analogue, for ramified coverings of Riemann surfaces, which is simpler in that only one kind of subgroup of G need be considered, rather than two. This was certainly familiar before Hilbert.

Let L/K be a finite extension of number fields, and let O_K and O_L be the corresponding ring of integers of K and L, respectively, which are defined to be the integral closure of the integers Z in the field in question.

Finally, let p be a non-zero prime ideal in O_K, or equivalently, a maximal ideal, so that the residue O_K/p is a field.

From the basic theory of one-dimensional rings follows the existence of a unique decomposition

of the ideal pO_L generated in O_L by p into a product of distinct maximal ideals P_j, with multiplicities e_j.

The field F = O_K/p naturally embeds into F_j = O_L/P_j for every j, the degree f_j = [O_L/P_j : O_K/p] of this residue field extension is called inertia degree of P_j over p.

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