Ichimura: Hilbert-Speiser number fields and Stickelberger ideals

Table of contents for this issue | Previous article | Next article

Humio Ichimura
Hilbert-Speiser number fields and Stickelberger ideals
Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), p. 589-607
Article: subscription required
Class. Math.: 11R33, 11R18

Résumé - Abstract

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 $(H_{p^{\infty}}^{'})$ when it satisfies $(H_{p^{n}}^{'})$ for all $n \geq 1$ . It is known that the rationals $ℚ$ satisfy $(H_{p^{\infty}}^{'})$ 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 $(H_{p^{\infty}}^{'})$ for a small prime number $p$ .

Bibliography

[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 2073643 | Zbl 1106.11039
[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 2558746 | Zbl pre05661599
[3] A. Fröhlich, Stickelberger without Gauss sums. Algebraic Number Fields (Durham Symposium, 1975, ed. A. Fröhlich), 589–607, Academic Press, London, 1977. MR 450227 | Zbl 0376.12002
[4] A. Fröhlich and M. J. Taylor, Algebraic Number Theory. Cambridge Univ. Press, Cambridge, 1993. MR 1215934 | Zbl 0744.11001
[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.
Cedram | MR 1305289 | Zbl 0822.11076
[6] C. Greither, Cyclic Galois Extensions of Commutative Rings. Springer, Berlin, 1992. MR 1222646 | Zbl 0788.13003
[7] D. Hilbert, The Theory of Algebraic Number Fields. Springer, Berlin, 1998. MR 1646901 | Zbl 0984.11001
[8] K. Horie, Ideal class groups of Iwasawa-theoretical abelian extensions over the rational field. J. London Math. Soc., 66 (2002), 257–275. MR 1920401 | Zbl 1011.11072
[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 2027901 | Zbl 1042.11074
[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.
Cedram | MR 2212125 | Zbl 1153.11335
[11] H. Ichimura, Stickelberger ideals and normal bases of rings of $p$ -integers. Math. J. Okayama Univ., 48 (2006), 9–20. MR 2291162 | Zbl pre05235572
[12] H. Ichimura, A class number formula for the $p$ -cyclotomic field. Arch. Math. (Basel), 87 (2006),539–545. MR 2283685 | Zbl 1120.11043
[13] H. Ichimura, Triviality of Stickelberger ideals of conductor $p$ . J. Math. Sci. Univ. Tokyo, 13 (2006), 617–628. MR 2306221 | Zbl pre05202850
[14] H. Ichimura, Hilbert-Speiser number fields for a prime $p$ inside the $p$ -cyclotomic field. J. Number Theory, 128 (2008), 858–864. MR 2400044 | Zbl 1167.11042
[15] H. Ichimura, On the parity of the class number of the $7^{n}$ -th cyclotomic field. Math. Slovaca, 59 (2009), 357–364. MR 2505815 | Zbl pre05564801
[16] H. Ichimura and H. Sumida-Takahashi, Stickelberger ideals of conductor $p$ and their application. J. Math. Soc. Japan, 58 (2006), 885–902.
Article | MR 2254415 | Zbl 1102.11059
[17] I. Kersten and J. Michalicek, $ℤ_{p}$ -extensions of complex multiplication fields. J. Number Theory, 32 (1989), 131–150. MR 1002468 | Zbl 0709.11057
[18] F. van der Linden, Class number computations of real abelian number fields. Math. Comp., 39 (1982), 639–707. MR 669662 | Zbl 0505.12010
[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 457403 | Zbl 0389.12005
[20] L. R. McCulloh, Galois module structure of elementary abelian extensions. J. Algebra, 82 (1983), 102–134. MR 701039 | Zbl 0508.12008
[21] L. R. McCulloh, Galois module structure of abelian extensions. J. Reine Angew. Math., 375/376 (1987), 259–306. MR 882300 | Zbl 0619.12008
[22] K. Rubin, Euler Systems. Princeton Univ. Press, Princeton, 2000. MR 1749177 | Zbl 0977.11001
[23] W. Sinnott, On the Stickelberger ideal and the circular units of an abelian field. Invent. Math., 62 (1980/81), 181–234. MR 595586 | Zbl 0465.12001
[24] L. C. Washington, Class numbers and $ℤ_{p}$ -extensions. Math. Ann., 214 (1975),177–193. MR 364182 | Zbl 0302.12007
[25] L. C. Washington, The non- $p$ -part of the class number in a cyclotomic $ℤ_{p}$ -extension. Invent. Math., 49 (1978), 87–97. MR 511097 | Zbl 0403.12007
[26] L. C. Washington, Introduction to Cyclotomic Fields (2nd. ed). Springer, New York, 1997. MR 1421575 | Zbl 0966.11047