Hilbert-Speiser number fields and Stickelberger ideals

Humio Ichimura[1]

  • [1] Faculty of Science, Ibaraki University Bunkyo 2-1-1, Mito, 310-8512, Japan

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 3, page 589-607
  • ISSN: 1246-7405

Abstract

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Let p be a prime number. We say that a number field F satisfies the condition ( H p n ′ ) when any abelian extension N / F of exponent dividing p n has a normal integral basis with respect to the ring of p -integers. We also say that F satisfies ( H p ∞ ′ ) when it satisfies ( H p n ′ ) for all n ≥ 1 . It is known that the rationals ℚ satisfy ( H p ∞ ′ ) for all prime numbers p . In this paper, we give a simple condition for a number field F to satisfy ( H p n ′ ) in terms of the ideal class group of K = F ( ζ p n ) and a “Stickelberger ideal” associated to the Galois group Gal ( K / F ) . As an application, we give a candidate of an imaginary quadratic field F which has a possibility of satisfying the very strong condition ( H p ∞ ′ ) for a small prime number p .

How to cite

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Ichimura, Humio. "Hilbert-Speiser number fields and Stickelberger ideals." Journal de Théorie des Nombres de Bordeaux 21.3 (2009): 589-607. <http://eudml.org/doc/10900>.

@article{Ichimura2009,
abstract = {Let $p$ be a prime number. We say that a number field $F$ satisfies the condition $(H_\{p^n\}^\{\prime\})$ when any abelian extension $N/F$ of exponent dividing $p^n$ has a normal integral basis with respect to the ring of $p$-integers. We also say that $F$ satisfies $(H_\{p^\{\infty \}\}^\{\prime\})$ when it satisfies $(H_\{p^n\}^\{\prime\})$ for all $n \ge 1$. It is known that the rationals $\mathbb\{Q\}$ satisfy $(H_\{p^\{\infty \}\}^\{\prime\})$ for all prime numbers $p$. In this paper, we give a simple condition for a number field $F$ to satisfy $(H_\{p^n\}^\{\prime\})$ in terms of the ideal class group of $K=F(\zeta _\{p^n\})$ and a “Stickelberger ideal” associated to the Galois group $\mbox \{\rm Gal\}(K/F)$. As an application, we give a candidate of an imaginary quadratic field $F$ which has a possibility of satisfying the very strong condition $(H_\{p^\{\infty \}\}^\{\prime\})$ for a small prime number $p$.},
affiliation = {Faculty of Science, Ibaraki University Bunkyo 2-1-1, Mito, 310-8512, Japan},
author = {Ichimura, Humio},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {normal integral basis; Stickelberger ideal},
language = {eng},
number = {3},
pages = {589-607},
publisher = {Université Bordeaux 1},
title = {Hilbert-Speiser number fields and Stickelberger ideals},
url = {http://eudml.org/doc/10900},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Ichimura, Humio
TI - Hilbert-Speiser number fields and Stickelberger ideals
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 3
SP - 589
EP - 607
AB - Let $p$ be a prime number. We say that a number field $F$ satisfies the condition $(H_{p^n}^{\prime})$ when any abelian extension $N/F$ of exponent dividing $p^n$ has a normal integral basis with respect to the ring of $p$-integers. We also say that $F$ satisfies $(H_{p^{\infty }}^{\prime})$ when it satisfies $(H_{p^n}^{\prime})$ for all $n \ge 1$. It is known that the rationals $\mathbb{Q}$ satisfy $(H_{p^{\infty }}^{\prime})$ for all prime numbers $p$. In this paper, we give a simple condition for a number field $F$ to satisfy $(H_{p^n}^{\prime})$ in terms of the ideal class group of $K=F(\zeta _{p^n})$ and a “Stickelberger ideal” associated to the Galois group $\mbox {\rm Gal}(K/F)$. As an application, we give a candidate of an imaginary quadratic field $F$ which has a possibility of satisfying the very strong condition $(H_{p^{\infty }}^{\prime})$ for a small prime number $p$.
LA - eng
KW - normal integral basis; Stickelberger ideal
UR - http://eudml.org/doc/10900
ER -

References

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