Hilbert-Speiser number fields and Stickelberger ideals

Ichimura, Humio

Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 589-607.

Soit $p$ un nombre premier. On dit qu’un corps de nombres $F$ satisfait la condition $\left(H_{p^n}^{\prime }\right)$ si toute extension abélienne $N/F$ d’exposant divisant $p^n$ possède une base normale d’entiers sur l’anneau des $p$-entiers. On dit aussi que $F$ satisfait la condition $\left(H_{p^{\infty }}^{\prime }\right)$ s’il satisfait $\left(H_{p^n}^{\prime }\right)$ pour tout $n\ge 1$. Il est bien connu que le corps des rationnels $\mathbb{Q}$ satisfait $\left(H_{p^{\infty }}^{\prime }\right)$ pour les nombres premiers $p$. Dans cet article, nous donnons une condition simple pour qu’un corps de nombres $F$ satisfasse $\left(H_{p^n}^{\prime }\right)$ en termes du groupe des classes d’idéaux de $K=F\left({\zeta }_{p^n}\right)$ et d’un “idéal de Stickelberger” associé au groupe de Galois $\mathrm{Gal}(K/F)$. Comme application, nous donnons un corps quadratique imaginaire qui pourait vérifier la condition très forte $\left(H_{p^{\infty }}^{\prime }\right)$ pour un petit nombre premier $p$.

Let $p$ be a prime number. We say that a number field $F$ satisfies the condition $\left(H_{p^n}^{\prime }\right)$ 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 $\left(H_{p^{\infty }}^{\prime }\right)$ when it satisfies $\left(H_{p^n}^{\prime }\right)$ for all $n\ge 1$. It is known that the rationals $\mathbb{Q}$ satisfy $\left(H_{p^{\infty }}^{\prime }\right)$ for all prime numbers $p$. In this paper, we give a simple condition for a number field $F$ to satisfy $\left(H_{p^n}^{\prime }\right)$ in terms of the ideal class group of $K=F\left({\zeta }_{p^n}\right)$ and a “Stickelberger ideal” associated to the Galois group $\mathrm{Gal}(K/F)$. As an application, we give a candidate of an imaginary quadratic field $F$ which has a possibility of satisfying the very strong condition $\left(H_{p^{\infty }}^{\prime }\right)$ for a small prime number $p$.

Classification : 11R33, 11R18

[1] J. Buhler, C. Pomerance and L. Robertson, Heuristics for class numbers of prime-power real cyclotomic fields. Fields Inst. Commun., 41 (2004), 149–157. | MR | Zbl

[2] I. Del Corso and L. P. Rossi, Normal integral bases for cyclic Kummer extensions. Preprint, 1.356.1706, Dipartimento di Matematica, Universita’ di Pisa. | MR

[3] A. Fröhlich, Stickelberger without Gauss sums. Algebraic Number Fields (Durham Symposium, 1975, ed. A. Fröhlich), 589–607, Academic Press, London, 1977. | MR | Zbl

[4] A. Fröhlich and M. J. Taylor, Algebraic Number Theory. Cambridge Univ. Press, Cambridge, 1993. | MR | Zbl

[5] E. J. Gómez Ayala, Bases normales d’entiers dans les extensions de Kummer de degré premier. J. Théor. Nombres Bordeaux, 6 (1994), 95–116. | Numdam | MR | Zbl

[6] C. Greither, Cyclic Galois Extensions of Commutative Rings. Springer, Berlin, 1992. | MR | Zbl

[7] D. Hilbert, The Theory of Algebraic Number Fields. Springer, Berlin, 1998. | MR | Zbl

[8] K. Horie, Ideal class groups of Iwasawa-theoretical abelian extensions over the rational field. J. London Math. Soc., 66 (2002), 257–275. | MR | Zbl

[9] H. Ichimura, On the ring of integers of a tame Kummer extension over a number field. J. Pure Appl. Algebra, 187 (2004), 169–182. | MR | Zbl

[10] H. Ichimura, On the ring of $p$-integers of a cyclic $p$-extension over a number field. J. Théor. Nombres Bordeaux, 17 (2005), 779–786. | Numdam | MR | Zbl

[11] H. Ichimura, Stickelberger ideals and normal bases of rings of $p$-integers. Math. J. Okayama Univ., 48 (2006), 9–20. | MR

[12] H. Ichimura, A class number formula for the $p$-cyclotomic field. Arch. Math. (Basel), 87 (2006),539–545. | MR | Zbl

[13] H. Ichimura, Triviality of Stickelberger ideals of conductor $p$. J. Math. Sci. Univ. Tokyo, 13 (2006), 617–628. | MR

[14] H. Ichimura, Hilbert-Speiser number fields for a prime $p$ inside the $p$-cyclotomic field. J. Number Theory, 128 (2008), 858–864. | MR | Zbl

[15] H. Ichimura, On the parity of the class number of the $7^n$-th cyclotomic field. Math. Slovaca, 59 (2009), 357–364. | MR

[16] H. Ichimura and H. Sumida-Takahashi, Stickelberger ideals of conductor $p$ and their application. J. Math. Soc. Japan, 58 (2006), 885–902. | MR | Zbl

[17] I. Kersten and J. Michalicek, ${\mathbb{Z}}_p$-extensions of complex multiplication fields. J. Number Theory, 32 (1989), 131–150. | MR | Zbl

[18] F. van der Linden, Class number computations of real abelian number fields. Math. Comp., 39 (1982), 639–707. | MR | Zbl

[19] L. R. McCulloh, A Stickelberger condition on Galois module structure for Kummer extensions of prime degree. Algebraic Number Fields (Durham Symposium, 1975, ed. A. Fröhlich), 561–588, Academic Press, London, 1977. | MR | Zbl

[20] L. R. McCulloh, Galois module structure of elementary abelian extensions. J. Algebra, 82 (1983), 102–134. | MR | Zbl

[21] L. R. McCulloh, Galois module structure of abelian extensions. J. Reine Angew. Math., 375/376 (1987), 259–306. | MR | Zbl

[22] K. Rubin, Euler Systems. Princeton Univ. Press, Princeton, 2000. | MR | Zbl

[23] W. Sinnott, On the Stickelberger ideal and the circular units of an abelian field. Invent. Math., 62 (1980/81), 181–234. | MR | Zbl

[24] L. C. Washington, Class numbers and ${\mathbb{Z}}_p$-extensions. Math. Ann., 214 (1975),177–193. | MR | Zbl

[25] L. C. Washington, The non-$p$-part of the class number in a cyclotomic ${\mathbb{Z}}_p$-extension. Invent. Math., 49 (1978), 87–97. | MR | Zbl

[26] L. C. Washington, Introduction to Cyclotomic Fields (2nd. ed). Springer, New York, 1997. | MR | Zbl