Unramified ideal

A prime ideal of an algebraic number field (cf. also Algebraic number; Number field) lying over a prime number such that the principal ideal has in a product decomposition into prime ideals of the form

where

and . More accurately, such an ideal is called absolutely unramified. In general, let be a Dedekind ring with field of fractions , let be a finite extension of and let be the integral closure of in (cf. Integral extension of a ring). A prime ideal of lying over an ideal of is unramified in the extension if

where are pairwise distinct prime ideals of , and . If all ideals are unramified, then one occasionally says that remains unramified in . For a Galois extension , an ideal of is unramified if and only if the decomposition group of in the Galois group is the same as the Galois group of the extension of the residue class field . In any finite extension of algebraic number fields all ideals except finitely many are unramified.

References

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