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 OK factorise as products of prime ideals of OL, 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 Pj, with multiplicities ej.
The field F = OK/p naturally embeds into Fj = OL/Pj for every j, the degree fj = [OL/Pj : OK/p] of this residue field extension is called inertia degree of Pj over p.
The multiplicity ej is called ramification index of Pj 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 Fj. 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 fj = ej = 1 for every j (and thus g = [L : K]), we say that p splits completely in L. If g = 1 and f1 = 1 (and so e1 = [L : K]), we say that p ramifies completely in L. Finally, if g = 1 and e1 = 1 (and so f1 = [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 prime avoidance lemma can be used to show the Galois group acts transitively on the Pj. 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 = fj and e = ej 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|/|DPj| for every j, where DPj, the decomposition group of Pj, is the subgroup of elements of G sending a given Pj 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 DPj is ef for every j.
This decomposition group contains a subgroup IPj, called inertia group of Pj, consisting of automorphisms of L/K that induce the identity automorphism on Fj. In other words, IPj is the kernel of reduction map . It can be shown that this map is surjective, and it follows that is isomorphic to DPj/IPj and the order of the inertia group IPj is e.
The theory of the Frobenius element goes further, to identify an element of DPj/IPj for given j which corresponds to the Frobenius automorphism in the Galois group of the finite field extension Fj /F. In the unramified case the order of DPj is f and IPj is trivial, so the Frobenius element is in this case an element of DPj, and thus also an element of G. For varying j, the groups DPj are conjugate subgroups inside G: Recalling that G acts transitively on the Pj, one checks that if maps Pj to P<sub>j</sub>, . Therefore, if G is an abelian group, the Frobenius element of an unramified prime P does not depend on which Pj we take. Furthermore, in the abelian case, associating an unramified prime of K to its Frobenius and extending multiplicatively defines a homomorphism from the group of unramified ideals of K into G. This map, known as the Artin map, is a crucial ingredient of class field theory, which studies the finite abelian extensions of a given number field K.
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.