Namely, let formula_2 be the ring of integers of an algebraic number field formula_3, and formula_4 a prime ideal of formula_2. For a field extension formula_6 we can consider the ring of integers formula_7 (which is the integral closure of formula_2 in formula_9), and the ideal formula_10 of formula_7. This ideal may or may not be prime, but for finite formula_12, it has a factorization into prime ideals:  where the formula_14 are distinct prime ideals of formula_7. Then formula_4 is said to ramify in formula_9 if formula_18 for some formula_19; otherwise it is . In other words, formula_4 ramifies in formula_9 if the ramification index formula_22 is greater than one for some formula_14. An equivalent condition is that formula_24 has a non-zero nilpotent element: it is not a product of finite fields. The analogy with the Riemann surface case was already pointed out by Richard Dedekind and Heinrich M. Weber in the nineteenth century. The ramification is encoded in formula_3 by the relative discriminant and in formula_9 by the relative different. The former is an ideal of formula_2 and is divisible by formula_4 if and only if some ideal formula_14 of formula_7 dividing formula_4 is ramified. The latter is an ideal of formula_7 and is divisible by the prime ideal formula_14 of formula_7 precisely when formula_14 is ramified. The ramification is tame when the ramification indices formula_22 are all relatively prime to the residue characteristic "p" of formula_4, otherwise wild. This condition is important in Galois module theory. A finite generically étale extension formula_38 of Dedekind domains is tame if and only if the trace formula_39 is surjective. The more detailed analysis of ramification in number fields can be carried out using extensions of the p-adic numbers, because it is a "local" question. In that case a quantitative measure of ramification is defined for Galois extensions, basically by asking how far the Galois group moves field elements with respect to the metric. A sequence of ramification groups is defined, reifying (amongst other things) "wild" (non-tame) ramification. This goes beyond the geometric analogue. In valuation theory, the ramification theory of valuations studies the set of extensions of a valuation of a field "K" to an extension field of "K".