Nagoya Mathematical Journal

On the unramified common divisor of discriminants of integers in a normal extension

Satomi Oka

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Abstract

Let $F$ be an algebraic number field of a finite degree, and $K$ be a normal extension over $F$ of a finite degree $n$. Let $\mathfrak{p}$ be a prime ideal of $F$ which is unramified in $K/F$, $\mathfrak{P}$ be a prime ideal of $K$ dividing $\mathfrak{p}$ such that $N_{K/F}\mathfrak{P} = \mathfrak{p}^f$, $n=fg$. Denote by $\delta(K/F)$ the greatest common divisor of discriminants of integers of $K$ with respect to $K/F$. Then, $\mathfrak{p}$divides $\delta(K/F)$ if and only if $\Sigma_{d|f} \mu(\frac fd)N\mathfrak{p}^d < n$.

Article information

Source
Nagoya Math. J., Volume 160 (2000), 181-186.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631505

Mathematical Reviews number (MathSciNet)
MR1804144

Zentralblatt MATH identifier
0967.11044

Subjects
Primary: 11R04: Algebraic numbers; rings of algebraic integers
Secondary: 11R21: Other number fields 11R29: Class numbers, class groups, discriminants

Citation

Oka, Satomi. On the unramified common divisor of discriminants of integers in a normal extension. Nagoya Math. J. 160 (2000), 181--186. https://projecteuclid.org/euclid.nmj/1114631505


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References

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