Tokyo Journal of Mathematics

The Capitulation Problem for Certain Cyclic Quartic Number Fields

Abdelmalek AZIZI, Idriss JERRARI, and Mohammed TALBI

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Abstract

Let $K$ be a cyclic quartic number field such that its 2-class group is of type $(2,4)$, $K_2^{(1)}$ be the Hilbert 2-class field of $K$, $K_2^{(2)}$ be the Hilbert 2-class field of $K_2^{(1)}$ and $G=\text{Gal}(K_2^{(2)}/K)$ be the Galois group of $K_2^{(2)}/K$. Our goal is to study the capitulation problem of 2-ideal classes of $K$ and to determine the structure of $G$.

Article information

Source
Tokyo J. Math., Volume 39, Number 2 (2016), 351-359.

Dates
First available in Project Euclid: 20 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1484903127

Digital Object Identifier
doi:10.3836/tjm/1484903127

Mathematical Reviews number (MathSciNet)
MR3599497

Zentralblatt MATH identifier
1381.11104

Subjects
Primary: 11R27: Units and factorization
Secondary: 11R37: Class field theory

Citation

AZIZI, Abdelmalek; JERRARI, Idriss; TALBI, Mohammed. The Capitulation Problem for Certain Cyclic Quartic Number Fields. Tokyo J. Math. 39 (2016), no. 2, 351--359. doi:10.3836/tjm/1484903127. https://projecteuclid.org/euclid.tjm/1484903127


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References

  • A. Azizi and M. Talbi, Capitulation des $2$-classes d'idéaux de certains corps biquadratiques cycliques, Acta Arithmetica 127 (2007), 231–248.
  • A. Azizi and M. Talbi, Capitulation dans certaines extensions non ramifiées de corps quartiques cycliques, Archivum Mathematicum (Brno), Tomus 44 (2008), 271–284.
  • A. Azizi and M. Taous, Capitulation des $2$-classes d'idéaux de type $(2,4)$, arXiv 30 Jan 2014.
  • P. Barrucand and H. Cohn, Note on primes of type $x^2 +32y^2$, class number and residuacity, J. Reine Angew. Math. 238 (1969), 67–70.
  • E. Benjamin, F. Lemmermeyer and C. Snyder, Real quadratic fields with abelian $2$-class field tower, J. Number Theory 73 (1998), 182–194.
  • E. Brown and C. J. Parry, The $2$-class group of certain biquadratic number fields I, J. Reine Angew. Math. 295 (1977), 61–71.
  • E. Brown and C. J. Parry, The $2$-class group of certain biquadratic number fields II, Pacific J. Math. 78, No. 1, (1978), 11–26.
  • H. Cohn, The explicit Hilbert $2$-cyclic class fields of $\mathbb{Q}(\sqrt{-p})$, J. Reine Angew. Math. 321 (1981), 64–77.
  • P. E. Conner and J. Hurrelbrink, Class Number Parity, Series in Pure Mathematic, World Scientific, 1988.
  • M. Ishida, The genus fields of algebraic number fields, Lecture notes in mathematics 555, Springer-Verlag, 1976.
  • P. Kaplan, Divisibilité par $8$ du nombre des classes des corps quadratiques dont le $2$-groupe des classes est cyclique, et réciprocité biquadratique, J. Math. Soc. Japan 25, No. $4$, (1973), 596–608.
  • P. Kaplan, Sur le $2$-groupe des classes d'idéaux des corps quadratiques, J. Reine Angew. Math. 283/284 (1976), 313–363.
  • R. Kučera, On the parity of the class number of biquadratic field, J. Number Theory 52 (1995), 43–52.
  • F. Lemmermeyer, Kuroda's class number formula, Acta Arith. 66(3) (1994), 245–260.
  • F. Lemmermeyer, On the $2$-class field tower of imaginary quadratic number fields, J. Théorie des Nombres de Bordeaux 6 (1994), 261–272.
  • M. Talbi, Capitulation des $2$-classes d'idéaux de certains corps quartiques de type $(2,2)$, thèse. Université Mohamed Premier. Oujda, 2008.
  • O. Taussky, A Remark on the Class Field Tower, J. London Math. Soc. 12 (1937), 82–85.
  • L. C. Washington, Introduction to cyclotomic fields, Graduate texts in mathematics 83 (1996).