# Unramified ideal

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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)
How to Cite This Entry:
Unramified ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unramified_ideal&oldid=49096
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article