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.
O K ↪ O L ↓ ↓ K ↪ L {\displaystyle {\begin{array}{ccc}O_{K}&\hookrightarrow &O_{L}\\\downarrow &&\downarrow \\K&\hookrightarrow &L\end{array}}}
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
p O L = ∏ j = 1 g P j e j {\displaystyle pO_{L}=\prod _{j=1}^{g}P_{j}^{e_{j}}}
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
[ L : K ] = ∑ j = 1 g e j f j . {\displaystyle [L:K]=\sum _{j=1}^{g}e_{j}f_{j}.}
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 G = Gal ( L / K ) {\displaystyle G=\operatorname {Gal} (L/K)} 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
p O L = ( ∏ j = 1 g P j ) e {\displaystyle pO_{L}=\left(\prod _{j=1}^{g}P_{j}\right)^{e}} . and
[ L : K ] = e f g . {\displaystyle [L:K]=efg.}
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