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)