Proceedings of the Japan Academy, Series A, Mathematical Sciences

Real abelian fields satisfying the Hilbert-Speiser condition for some small primes $p$

Humio Ichimura

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Abstract

For a prime number $p$, we say that a number field $F$ satisfies the Hilbert-Speiser condition $(H_{p})$ if each tame cyclic extension $N/F$ of degree $p$ has a normal integral basis. In this note, we determine the real abelian number fields satisfying $(H_{p})$ for odd prime numbers $p$ with $h(\mathbf{Q}(\sqrt{-p}))=1$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 1 (2016), 19-22.

Dates
First available in Project Euclid: 28 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.pja/1451330562

Digital Object Identifier
doi:10.3792/pjaa.92.19

Mathematical Reviews number (MathSciNet)
MR3447745

Zentralblatt MATH identifier
06586130

Subjects
Primary: 11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10] 11R18: Cyclotomic extensions

Keywords
Hilbert-Speiser number fields real abelian fields

Citation

Ichimura, Humio. Real abelian fields satisfying the Hilbert-Speiser condition for some small primes $p$. Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 1, 19--22. doi:10.3792/pjaa.92.19. https://projecteuclid.org/euclid.pja/1451330562


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References

  • J. Brinkhuis, Normal integral bases and complex conjugation, J. Reine Angew. Math. 375/376 (1987), 157–166.
  • N. P. Byott, J. E. Carter, C. Greither and H. Johnston, On the restricted Hilbert-Speiser and Leopoldt properties, Illinois J. Math. 55 (2011), no. 2, 623–639.
  • 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, Arch. Math. (Basel) 83 (2004), no. 6, vi–vii.
  • D. A. Cox, Primes of the form $x^{2} + ny^{2}$, A Wiley-Interscience Publication, Wiley, New York, 1989.
  • C. Greither and H. Johnston, On totally real Hilbert-Speiser fields of type $C_{p}$, Acta Arith. 138 (2009), no. 4, 329–336.
  • C. Greither, D. R. Replogle, K. Rubin and A. Srivastav, Swan modules and Hilbert-Speiser number fields, J. Number Theory 79 (1999), no. 1, 164–173.
  • H. Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie Verlag, Berlin, 1952.
  • T. Herreng, Sur les corps de Hilbert-Speiser, J. Théor. Nombres Bordeaux 17 (2005), no. 3, 767–778.
  • 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.
  • H. Ichimura, Normal integral bases and ray class groups, Acta Arith. 114 (2004), no. 1, 71–85.
  • H. Ichimura, Hilbert-Speiser number fields and the complex conjugation, J. Math. Soc. Japan 62 (2010), no. 1, 83–94.
  • H. Ichimura and H. Sumida-Takahashi, Imaginary quadratic fields satisfying the Hilbert-Speiser type condition for a small prime $p$, Acta Arith. 127 (2007), no. 2, 179–191.
  • H. Ichimura and H. Sumida-Takahashi, On Hilbert-Speiser type imaginary quadratic fields, Acta Arith. 136 (2009), no. 4, 385–389.
  • S. Mäki, The determination of units in real cyclic sextic fields, Lecture Notes in Mathematics, 797, Springer, Berlin, 1980.
  • L. R. McCulloh, Galois module structure of elementary abelian extensions, J. Algebra 82 (1983), no. 1, 102–134.
  • W. Sinnott, On the Stickelberger ideal and the circular units of an abelian field, Invent. Math. 62 (1980/81), no. 2, 181–234.
  • L. C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, 83, Springer, New York, 1997.
  • K. Yamamura, The determination of the imaginary abelian number fields with class number one, Math. Comp. 62 (1994), no. 206, 899–921.
  • K. Yamamura, http://tnt.math.se.tmu.ac.jp/pub/ac11/rcn/composite/
  • Y. Yoshimura, Abelian number fields satisfying the Hilbert-Speiser condition at $p=2$ or 3, Tokyo J. Math. 32 (2009), no. 1, 229–235.
  • K. Yoshino and M. Hirabayashi, On the relative class number of the imaginary abelian number field I, Memoirs of the College of Liberal Arts, Kanazawa Medical University 9 (1981), 5–53.
  • K. Yoshino and M. Hirabayashi, On the relative class number of the imaginary abelian number field II, Memoirs of the College of Liberal Arts, Kanazawa Medical University 10 (1982), 33–81.

Corrections

  • Humio Ichimura. Corrigendum to “Real abelian fields satisfying the Hilbert-Speiser condition for some small primes $p$”. Proc. Japan Acad. Ser. A Math. Sci., Volume 95, Number 7 (2019), 80-82.