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.

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 Transitivity on the P j . That is, the prime ideal factors of P in L form a single orbit (group theory) 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 is 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 j O 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 j O 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.