Note on imaginary quadratic fields satisfying the Hilbert-Speiser condition at a prime p
Humio Ichimura
Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 83, Number 6
(2007), 88-91.
Abstract
Let $p$ be a prime number. A number field $F$ satisfies the condition $(H_p)$ when any tame cyclic extention $N/F$ of degree $p$ has a normal integral basis. For the case $p=2$, it is shown by Mann that $F$ satisfies $(H_2)$ only when $h_F=1$ where $h_F$ is the class number of $F$. We prove that if an imaginary quadratic field $F$ satisfies $(H_p)$ for some $p$, then $h_F=1$.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.pja/1188405577
Mathematical Reviews number (MathSciNet): MR2355504
Digital Object Identifier: doi:10.3792/pjaa.83.88
Zentralblatt MATH identifier: 1157.11044
References
J. E. Carter, Normal integral bases in quadratic and cyclic cubic extensions of quadratic fields, Arch. Math. (Basel) 81 (2003), no. 3, 266--271: Erratum, ibid., 83 (2004), no.6, vi-vii.
Mathematical Reviews (MathSciNet): MR2013257
Digital Object Identifier: doi:10.1007/s00013-003-0821-1
Zentralblatt MATH: 1050.11097
A. Fröhlich %%!! and M. J. Taylor, Algebraic number theory, Cambridge Univ. Press, Cambridge, 1993.
Mathematical Reviews (MathSciNet): MR1215934
C. Greither et al., Swan modules and Hilbert-Speiser number fields, J. Number Theory 79 (1999), no. 1, 164--173.
Mathematical Reviews (MathSciNet): MR1718724
Digital Object Identifier: doi:10.1006/jnth.1999.2425
Zentralblatt MATH: 0941.11044
H. Ichimura, Note on the ring of integers of a Kummer extension of prime degree. V, Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 6, 76--79.
Mathematical Reviews (MathSciNet): MR1913934
Project Euclid: euclid.pja/1148392678
H. Ichimura, Normal integral bases and ray class groups, Acta Arith. 114 (2004), no. 1, 71--85.
Mathematical Reviews (MathSciNet): MR2067873
H. Ichimura, Normal integral bases and ray class groups. II, Yokohama Math. J. 53 (2006), no. 1, 75--81.
Mathematical Reviews (MathSciNet): MR2295067
H. Ichimura and F. Kawamoto, Normal integral basis and ray class group modulo 4, Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), no. 9, 139--141.
Mathematical Reviews (MathSciNet): MR2022056
Project Euclid: euclid.pja/1116443729
H. Ichimura and H. Sumida-Takahashi, Imaginary quadratic fields satisfying the Hilbert-Speiser type condition for a small prime $p$, Acta Arith., 127 (2007), 179--191.
Mathematical Reviews (MathSciNet): MR2289983
H. B. Mann, On integral bases, Proc. Amer. Math. Soc. 9 (1958), 167--172.
Mathematical Reviews (MathSciNet): MR93502
Digital Object Identifier: doi:10.2307/2033418
JSTOR: links.jstor.org
L. R. McCulloh, Galois module structure of elementary abelian extensions, J. Algebra 82 (1983), no. 1, 102--134.
Mathematical Reviews (MathSciNet): MR701039
Digital Object Identifier: doi:10.1016/0021-8693(83)90176-X
Zentralblatt MATH: 0508.12008
D. R. Replogle, Kernel groups and nontrivial Galois module structure of imaginary quadratic fields, Rocky Mountain J. Math. 34 (2004), no. 1, 309--320.
Mathematical Reviews (MathSciNet): MR2061133
Digital Object Identifier: doi:10.1216/rmjm/1181069907
Project Euclid: euclid.rmjm/1181069907
Zentralblatt MATH: 1064.11076
H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1967), 1--27.
Mathematical Reviews (MathSciNet): MR222050
Digital Object Identifier: doi:10.1307/mmj/1028999653
Project Euclid: euclid.mmj/1028999653
L. C. Washington, Introduction to cyclotomic fields, Second edition, Springer, New York, 1997.
Mathematical Reviews (MathSciNet): MR1421575
Zentralblatt MATH: 0966.11047
Proceedings of the Japan Academy, Series A, Mathematical Sciences
