[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 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. Also the Frobenius element is in this case an element of DPj (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