Unramified ideal

2020 Mathematics Subject Classification: Primary: 11S15 [MSN][ZBL]

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

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

where

$$ \mathfrak P _ {1} = \mathfrak P \ \textrm{ and } \ \ \mathfrak P _ {2} \dots \mathfrak P _ {s} \neq \mathfrak P , $$

and $ e _ {1} = 1 $. More accurately, such an ideal is called absolutely unramified. In general, let $ A $ be a Dedekind ring with field of fractions $ k $, let $ K $ be a finite extension of $ k $ and let $ B $ be the integral closure of $ A $ in $ K $( cf. Integral extension of a ring). A prime ideal $ \mathfrak P $ of $ B $ lying over an ideal $ \mathfrak Y $ of $ A $ is unramified in the extension $ K / k $ if

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

where $ \mathfrak P _ {1} \dots \mathfrak P _ {s} $ are pairwise distinct prime ideals of $ B $, $ \mathfrak P _ {1} = \mathfrak P $ and $ e _ {1} = 1 $. If all ideals $ \mathfrak P _ {1} \dots \mathfrak P _ {s} $ are unramified, then one occasionally says that $ \mathfrak Y $ remains unramified in $ K / k $. For a Galois extension $ K / k $, an ideal $ \mathfrak P $ of $ B $ is unramified if and only if the decomposition group of $ \mathfrak P $ in the Galois group $ G ( K / k ) $ is the same as the Galois group of the extension of the residue class field $ ( B/ \mathfrak P ) / ( A/ \mathfrak Y ) $. 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)