Ramified prime ideal

A prime ideal in a Dedekind ring $ A $ which divides the discriminant of a finite separable extension $ K/k $, where $ k $ is the field of fractions of $ A $. Such ideals are the only ideals that are ramified in the extension $ K/k $. A prime ideal $ \mathfrak p $ of a ring $ A $ is ramified in $ K/k $ if the following product representation holds in the integral closure $ B $ of $ A $ in the field $ K $:

$$ \mathfrak p B = \ \mathfrak P _ {1} ^ {e _ {1} } \dots \mathfrak P _ {s} ^ {e _ {s} } , $$

where $ \mathfrak P _ {1} \dots \mathfrak P _ {s} $ are prime ideals in $ B $ and at least one of the numbers $ e _ {i} $ is greater than 1. The number $ e _ {i} $ is called the ramification index of $ \mathfrak P _ {i} $ over $ \mathfrak p $.

If $ K/k $ is a Galois extension with Galois group $ G ( K/k) $, then $ e _ {1} = \dots = e _ {s} $ and $ e _ {i} $ is precisely the order of the inertia subgroup $ T ( \mathfrak P _ {i} ) $ of $ \mathfrak P _ {i} $ in $ G ( K/k) $:

$$ T ( \mathfrak P _ {i} ) = \ \{ {\sigma \in G ( K/k) } : { \sigma a - a \in \mathfrak P _ {i} \ \textrm{ for } \textrm{ all } \ a \in B } \} . $$

Other, more refined, characteristics of the ramification are given by the higher ramification groups $ T ( \mathfrak P _ {i} ) _ {n} \subset T ( \mathfrak P _ {i} ) $, $ n = 1, 2 \dots $ defined as follows:

$$ T ( \mathfrak P _ {i} ) _ {n} = \ \{ {\sigma \in G ( K/k) } : { \sigma a - a \in \mathfrak P _ {i} ^ {n + 1 } \ \textrm{ for } \textrm{ all } a \in B } \} . $$

Let $ A = \mathbf Z $; by Minkowski's theorem, for any finite extension of the field $ \mathbf Q $ of rational numbers there exists a ramified prime ideal. This is not true for arbitrary algebraic number fields: If the field $ k $ has class number $ h > 1 $, i.e. has a non-trivial ideal class group, then there exist unramified extensions over $ k $, i.e. extensions having no ramified prime ideal. An example of such an extension is the Hilbert class field of the field $ k $; e.g., the field $ \mathbf Q ( \sqrt 5 , \sqrt - 5 ) $ is the Hilbert class field of $ \mathbf Q ( \sqrt - 5 ) $ and is unramified over $ \mathbf Q ( \sqrt - 5 ) $.

References

[1]	Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1987) (Translated from Russian) (German translation: Birkhäuser, 1966)
[2]	J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)
[3]	S. Lang, "Algebraic number theory" , Addison-Wesley (1970)