Splitting of Prime Ideals in Galois Extensions - Computing The Factorisation

Computing The Factorisation

Suppose that we wish to determine the factorisation of a prime ideal P of O_K into primes of O_L. We will assume that the extension L/K is a finite separable extension; the extra hypothesis of normality in the definition of Galois extension is not necessary.

The following procedure (Neukirch, p47) solves this problem in many cases. The strategy is to select an integer θ in O_L so that L is generated over K by θ (such a θ is guaranteed to exist by the primitive element theorem), and then to examine the minimal polynomial H(X) of θ over K; it is a monic polynomial with coefficients in O_K. Reducing the coefficients of H(X) modulo P, we obtain a monic polynomial h(X) with coefficients in F, the (finite) residue field O_K/P. Suppose that h(X) factorises in the polynomial ring F as

where the h_j are distinct monic irreducible polynomials in F. Then, as long as P is not one of finitely many exceptional primes (the precise condition is described below), the factorisation of P has the following form:

where the Q_j are distinct prime ideals of O_L. Furthermore, the inertia degree of each Q_j is equal to the degree of the corresponding polynomial h_j, and there is an explicit formula for the Q_j:

In the Galois case, the inertia degrees are all equal, and the ramification indices e₁ = ... = e_n are all equal.

The exceptional primes, for which the above result does not necessarily hold, are the ones not relatively prime to the conductor of the ring O_K. The conductor is defined to be the ideal

it measures how far the order O_K is from being the whole ring of integers (maximal order) O_L.

A significant caveat is that there exist examples of L/K and P such that there is no available θ that satisfies the above hypotheses (see for example ). Therefore the algorithm given above cannot be used to factor such P, and more sophisticated approaches must be used, such as that described in.