Editing Ramification (mathematics) (section)

==In algebraic number theory==
=== In algebraic extensions of the rational numbers ===
{{see also|Splitting of prime ideals in Galois extensions}}

Ramification in [[algebraic number theory]] means a prime ideal factoring in an extension so as to give some repeated prime ideal factors. Namely, let <math>\mathcal{O}_K</math> be the [[ring of integers]] of an [[algebraic number field]] <math>K</math>, and <math>\mathfrak{p}</math> a [[prime ideal]] of <math>\mathcal{O}_K</math>. For a field extension <math>L/K</math> we can consider the
ring of integers <math>\mathcal{O}_L</math> (which is the [[integral closure]] of <math>\mathcal{O}_K</math> in <math>L</math>), and the ideal <math>\mathfrak{p}\mathcal{O}_L</math> of <math>\mathcal{O}_L</math>. This ideal may or may not be prime, but for finite <math>[L:K]</math>, it has a factorization into prime ideals:

:<math>\mathfrak{p}\cdot \mathcal{O}_L = \mathfrak{p}_1^{e_1}\cdots\mathfrak{p}_k^{e_k}</math>

where the <math>\mathfrak{p}_i</math> are distinct prime ideals of <math>\mathcal{O}_L</math>. Then <math>\mathfrak{p}</math> is said to '''ramify''' in <math>L</math> if <math>e_i > 1</math> for some <math>i</math>; otherwise it is '''{{visible anchor|unramified}}'''. In other words, <math>\mathfrak{p}</math> ramifies in <math>L</math> if the '''ramification index''' <math>e_i</math> is greater than one for some <math>\mathfrak{p}_i</math>. An equivalent condition is that <math>\mathcal{O}_L/\mathfrak{p}\mathcal{O}_L</math> has a non-zero [[nilpotent]] element: it is not a product of [[finite field]]s. 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 <math>K</math> by the [[relative discriminant]] and in <math>L</math> by the [[relative different]].  The former is an ideal of <math>\mathcal{O}_K</math> and is divisible  by <math>\mathfrak{p}</math> if and only if some ideal <math>\mathfrak{p}_i</math> of <math>\mathcal{O}_L</math> dividing <math>\mathfrak{p}</math> is ramified.  The latter is an ideal of <math>\mathcal{O}_L</math> and is divisible by the prime ideal <math>\mathfrak{p}_i</math> of <math>\mathcal{O}_L</math> precisely when <math>\mathfrak{p}_i</math> is ramified.

The ramification is '''tame''' when the ramification indices <math>e_i</math> are all relatively prime to the residue characteristic ''p'' of <math>\mathfrak{p}</math>, otherwise '''wild'''. This condition is important in [[Galois module]] theory. A finite generically étale extension <math>B/A</math> of [[Dedekind domain]]s is tame if and only if the trace <math>\operatorname{Tr}: B \to A</math> is surjective.

===In local fields===
{{main|Ramification of local fields}}

The more detailed analysis of ramification in number fields can be carried out using extensions of the [[p-adic number]]s, because it is a ''local'' question. In that case a quantitative measure of ramification is defined for [[Galois extension]]s, basically by asking how far the [[Galois group]] moves field elements with respect to the metric. A sequence of [[ramification group]]s is defined, reifying (amongst other things) ''wild'' (non-tame) ramification. This goes beyond the geometric analogue.