Splitting of prime ideals in Galois extensions

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.
Definitions.
Let "L"/"K" be a finite extension of number fields, and let "OK" and "OL" 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 "OK", or equivalently, a maximal ideal, so that the residue "OK"/"p" is a field.
From the basic theory of one-dimensional rings follows the existence of a unique decomposition
of the ideal "pOL" generated in "OL" by "p" into a product of distinct maximal ideals "P""j", with multiplicities "e""j".
The field "F" = "OK"/"p" naturally embeds into "F""j" = "OL"/"P""j" for every "j", the degree "f""j" = ["OL"/"P""j" : "OK"/"p"] of this residue field extension is called inertia degree of "P""j" over "p".
The multiplicity "e""j" is called ramification index of "P""j" over "p". If it is bigger than 1 for some "j", the field extension "L"/"K" is called ramified at "p" (or we say that "p" ramifies in "L", or that it is ramified in "L"). Otherwise, "L"/"K" is called unramified at "p". If this is the case then by the Chinese remainder theorem the quotient "OL"/"pOL" is a product of fields "F""j". The extension "L"/"K" is ramified in exactly those primes that divide the relative discriminant, hence the extension is unramified in all but finitely many prime ideals.
Multiplicativity of ideal norm implies
If "f""j" = "e""j" = 1 for every "j" (and thus "g" = ["L" : "K"]), we say that "p" splits completely in "L". If "g" = 1 and "f""1" = 1 (and so "e""1" = ["L" : "K"]), we say that "p" ramifies completely in "L". Finally, if "g" = 1 and "e""1" = 1 (and so "f""1" = ["L" : "K"]), we say that "p" is inert in "L".
The Galois situation.
In the following, the extension "L"/"K" is assumed to be a Galois extension. Then the Galois group formula_4 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 and the unique factorisation theorem, it follows that "f" = "f""j" and "e" = "e""j" are independent of "j"; something that certainly need not be the case for extensions that are not Galois. The basic relations then read
and
The relation above shows that ["L" : "K"]/"ef" equals the number "g" of prime factors of "p" in "OL". By the orbit-stabilizer formula this number is also equal to |"G"|/|"D""P""j"| for every "j", where "D""P""j", the decomposition group of "P""j", is the subgroup of elements of "G" sending a given "P""j" to itself. Since the degree of "L"/"K" and the order of "G" are equal by basic Galois theory, it follows that the order of the decomposition group "D""P""j" is "ef" for every "j".
This decomposition group contains a subgroup "I""P""j", called inertia group of "P""j", consisting of automorphisms of "L"/"K" that induce the identity automorphism on "F""j". In other words, "I""P""j" is the kernel of reduction map formula_7. It can be shown that this map is surjective, and it follows that formula_8 is isomorphic to "D""P""j"/"I""P""j" and the order of the inertia group "I""P""j" is "e".
The theory of the Frobenius element goes further, to identify an element of "D""P""j"/"I""P""j" for given "j" which corresponds to the Frobenius automorphism in the Galois group of the finite field extension "F""j" /"F". In the unramified case the order of "D""P""j" is "f" and "I""P""j" is trivial. Also the Frobenius element is in this case an element of "D""P""j" (and thus also element of "G").
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, and there aren't many quadratic fields with unique factorization — 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]:
The ramification index here is therefore "e" = 2. The residue field is
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
for any integers "a" and "b", as formula_12 .
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:
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,
which are both isomorphic to the finite field with 13 elements. The Frobenius element is the trivial automorphism; this means that
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
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
for any integers "a" and "b".
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 (Neukirch, p. 47) 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"["X"] as
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:
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":
where "h""j" denotes here a lifting of the polynomial "h""j" to "K"["X"].
In the Galois case, the inertia degrees are all equal, and the ramification indices "e"1 = ... = "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
it measures how far the order "O""K"[θ] 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.
An example.
Consider again the case of the Gaussian integers. We take θ to be the imaginary unit "i", with minimal polynomial "H"("X") = "X"2 + 1. Since Z[formula_22] is the whole ring of integers of Q(formula_22), 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 polynomial "X"2 + 1 modulo 2:
Therefore, there is only one prime factor, with inertia degree 1 and ramification index 2, and it is given by
The next case is for "P" = ("p") for a prime "p" ≡ 3 mod 4. For concreteness we will take "P" = (7). The polynomial "X"2 + 1 is irreducible modulo 7. Therefore, there is only one prime factor, with inertia degree 2 and ramification index 1, and it is given by
The last case is "P" = ("p") for a prime "p" ≡ 1 mod 4; we will again take "P" = (13). This time we have the factorisation
Therefore, there are "two" prime factors, both with inertia degree and ramification index 1. They are given by
and