Splitting of prime ideals in Galois extensions

In mathematics, the interplay between the Galois group G of a Galois extension of number fields L/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.

Technical overview

The most basic fact of the theory is that if we write

PO_L = Π P_j^e(j)

as a product of distinct prime ideals P_j O_L, with multiplicities e(j), then G acts transitively on the P_j. That is, the prime ideal factors of P in L form a single orbit under the automorphisms of L over K. From this it follows at once, because there is unique prime factorisation into prime ideals, that e(j) = e is independent of j; something that certainly need not be the case for extensions that are not Galois.

The basic relation therefore reads

PO_L = (Π P_j)^e

and for all but the finite number of ramified P we must have e = 1. Considering first the unramified case, the quotient

O_L/PO_L

will be a product of fields

F_j = O_L/P_jO_L

and these are all isomorphic, say to the finite field F′, containing

F = O_K/P

By a counting argument we must have

[L:K]/[F′:F]

equal to the number of prime factors of P in O_L. By the orbit-stabilizer formula we must also have this number equal to

|G|/|D|

where by definition D, the decomposition group of P, is the subgroup of G sending a given P_j to itself. That is, since the degree of L/K and the order of G are equal by basic Galois theory, the order of the decomposition group D is the degree of the residue field extension F′/F. The theory of the Frobenius element goes further, to identify an element of D, for j given, which generated the Galois group of the finite field extension.

In the ramified case, there is the further phenomenon of inertia: the index e is interpreted as the extent to which elements of G are not seen in the Galois groups of any of the residue field extensions. Each decomposition group D, for a given P_j, contains an inertia group I consisting of the g in G that send P_j to itself, but induce the identity automorphism on

F_j = O_L/P_jO_L.

In the geometric analogue, for complex manifolds or algebraic geometry over an algebraically closed field, the concepts of decomposition group and inertia group coincide. There, given a Galois ramified cover, all but finitely many points have the same number of preimages.

The splitting of primes in extensions that are not Galois may be studied by using a splitting field initially, i.e. a Galois extension that is somewhat larger. For example cubic fields usually are 'regulated' by a degree 6 field containing them.

Example — the Gaussian integers

This section describes the splitting of prime ideals in the field extension Q(i)/Q. That is, we take K = Q and L = Q(i), so O_K is simply Z, and O_L = Z[i] is the ring of Gaussian integers. Although this case is far from representative — after all, Z[i] has unique factorisation — it exhibits many of the features of the theory.

Writing G for the Galois group of Q(i)/Q, and σ for the complex conjugation automorphism in G, there are three cases to consider.

The prime p = 2

The prime 2 of Z ramifies in Z[i]:

(2) = (1+i)²,

so the ramification index here is e = 2. The residue field is

O_L / (1+i)O_L

which is the finite field with two elements. The decomposition group must be equal to all of G, since there is only one prime of Z[i] above 2. The inertia group is also all of G, since

a + bi ≡ a − bi

modulo (1+i), for any integers a and b.

In fact, 2 is the only prime that ramifies in Z[i], since every prime that ramifies must divide the discriminant of Z[i], which is −4.

Primes p ≡ 1 mod 4

Any prime p ≡ 1 mod 4 splits into two distinct prime ideals in Z[i]; this is a manifestation of Fermat's theorem on sums of two squares. For example,

(13) = (2 + 3i)(2 − 3i).

The decomposition groups in this case are both the trivial group {1}; indeed the automorphism σ switches the two primes (2 + 3i) and (2 − 3i), so it cannot be in the decomposition group of either prime. The inertia group, being a subgroup of the decomposition group, is also the trivial group. There are two residue fields, one for each prime,

O_L / (2 ± 3i)O_L,

which are both isomorphic to the finite field with 13 elements. The Frobenius element is the trivial automorphism; this means that

(a + bi)¹³ ≡ a + bi

modulo (2 ± 3i), for any integers a and b.

Primes p ≡ 3 mod 4

Any prime p ≡ 3 mod 4 remains inert in Z[i]; that is, it does not split. For example, (7) remains prime in Z[i]. In this situation, the decomposition group is all of G, again because there is only one prime factor. However, this situation differs from the p = 2 case, because now σ does not act trivially on the residue field

O_L / (7)O_L,

which is the finite field with 7² = 49 elements. For example, the difference between 1 + i and σ(1 + i) = 1 − i is 2i, which is certainly not divisible by 7. Therefore the inertia group is the trivial group {1}. The Galois group of this residue field over the subfield Z/7Z has order 2, and is generated by the image of the Frobenius element. The Frobenius is none other than σ; this means that

(a + bi)⁷ ≡ a − bi

modulo 7, for any integers a and b.

Summary

Prime in Z	How it splits in Z[i]	Inertia group	Decomposition group
2	Ramifies with index 2	G	G
p ≡ 1 mod 4	Splits into two distinct factors	1	1
p ≡ 3 mod 4	Remains inert	1	G

Computing the factorisation

Suppose that we wish to determine the factorisation of a prime ideal P of O_K into primes of O_L. The following procedure (see Neukirch, p47) solves this problem in many cases. We need only 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 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[X] as

h(X)=h_{1}(X)^{e_{1}}\cdots h_{n}(X)^{e_{n}},

where the h_j are distinct monic irreducible polynomials in F[X]. 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:

PO_{L}=Q_{1}^{e_{1}}\cdots Q_{n}^{e_{n}},

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 f_j, and there is an explicit formula for the Q_j:

Q_{j}=PO_{L}+h_{j}(\theta )O_{L}.

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[θ]. The conductor is defined to be the ideal

\{y\in O_{L}:yO_{L}\subseteq O_{K}[\theta ]\};

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

An example

Consider again the case of the Gaussian integers. We take θ to be the imaginary unit i, with minimal polynomial H(X) = X² + 1. Since Z[i] is the whole ring of integers of Q(i), the conductor is the unit ideal, so there are no exceptional primes.

For P = (2), we need to work in the field Z/(2)Z, which amounts to factorising the polyomial X² + 1 modulo 2:

X^{2}+1=(X+1)^{2}{\pmod {2}}.

Therefore there is only one prime factor, with inertia degree 1 and ramification index 2, and it is given by

Q=(2)\mathbf {Z} [i]+(i+1)\mathbf {Z} [i]=(1+i)\mathbf {Z} [i].

The next case is for P = (p) for a prime p ≡ 3 mod 4. For concreteness we will take P = (7). The polynomial X² + 1 is irreducible modulo 7. Therefore there is only prime factor, with inertia degree 2 and ramification index 1, and it is given by