Home Mathematics On the 2-rank and 4-rank of the class group of some real pure quartic number fields
Article
Licensed
Unlicensed Requires Authentication

On the 2-rank and 4-rank of the class group of some real pure quartic number fields

  • Mbarek Haynou and Mohammed Taous EMAIL logo
Published/Copyright: April 14, 2021
Become an author with De Gruyter Brill

Abstract

Let K = ℚ(pd24) be a real pure quartic number field and k = ℚ(p) its real quadratic subfield, where p ≡ 5 (mod 8) is a prime integer and d an odd square-free integer coprime to p. In this work, we calculate r2(K), the 2-rank of the class group of K, in terms of the number of prime divisors of d that decompose or remain inert in ℚ(p), then we will deduce forms of d satisfying r2(K) = 2. In the last case, the 4-rank of the class group of K is given too.

MSC 2010: 11R11; 11R16; 11R29; 11R37
  1. (Communicated by István Gaál)

Acknowledgement

We would like to thank the unknown referee for his/her several helpful suggestions and for calling our attention to the missing details.

References

[1] Azizi, A.—Taous, M.: Determination des corps k = k=Q(d,i)dont les 2-groupes de classes sont de type (2, 4) ou (2, 2, 2), Rend. Istit. Mat. Univ. Trieste 40 (2008), 93–116.Search in Google Scholar

[2] Azizi, A.—Mouhib, A.: Sur le rang du 2-groupe de classes deQ(m,d)oú m = 2 ou un premier p ≡ 1 (mod 4), Trans. Amer. Math. Soc. 353(7) (2001), 2741–2752.10.1090/S0002-9947-01-02753-2Search in Google Scholar

[3] Azizi, A.—Mouhib, A.: Le 2-rang du groupe de classes de certains corps biquadratiques et applications, Internat. J. Math. 15(2) (2004), 169–182.10.1142/S0129167X04002235Search in Google Scholar

[4] Azizi, A.—Taous, M.—Zekhnini, A.: On the unit index of some real biquadratic number fields, Turk. J. Math. 42 (2018), 703–715.10.3906/mat-1704-63Search in Google Scholar

[5] Brown, E.—Parry, Ch. J.: The 2-class group of certain biquadratic number fields, J. Reine Angew. Math. 295 (1977), 61–71.10.1515/crll.1977.295.61Search in Google Scholar

[6] Brown, E.—Parry, Ch. J.: The 2-class group of certain biquadratic number fields II, Pacific J. Math. 78(1) (1978), 61–71.10.2140/pjm.1978.78.11Search in Google Scholar

[7] Chevalley, C.: Sur la théorie du corps de classes dans les corps finis et les corps locaux, J. Fac. Sci. Univ. Tokyo 2 (1933), 365–476.Search in Google Scholar

[8] Gras, G.: Class Field Theory, from Theory to Practice, Springer Verlag, 2003.10.1007/978-3-662-11323-3Search in Google Scholar

[9] Hymo, J. A.—Parry, C. J.: On relative integral bases for pure quartic fields, Indian J. Pure Appl. Math. 23 (1992), 359–376.Search in Google Scholar

[10] Lemmermeyer, F.: Reciprocity Laws. From Euler to Eisenstein, Springer Monographs in Math., 2000.10.1007/978-3-662-12893-0Search in Google Scholar

[11] Parry, C. J.: Pure quartic number fields whose class numbers are even, J. Reine Angew. Math. 264 (1975), 102–112.10.1515/crll.1975.272.102Search in Google Scholar

[12] Parry, C. J.: A genus theory for quartic fields, J. Reine Angew. Math. 314 (1980), 40–71.10.1515/crll.1980.314.40Search in Google Scholar

[13] The PARI Group.: PARI/GP version 2-11-3, Univ. Bordeaux, 2020, http://pari.math.u-bordeaux.fr.Search in Google Scholar

[14] Qin, Y.: The generalized Rédei-matrix, Math. Z. 261 (2009), 23–37.10.1007/s00209-008-0311-zSearch in Google Scholar

[15] Taous, M.: Capitulation des 2-classes d’idéaux de certains corpsk=Q(d,i)de type (2, 4), thèse, Université. Mohammed Premier Faculté des Science, Oujda. 2008.Search in Google Scholar

[16] Williams, H. C.: The quadratic character of a certain quadratic surds, Util. Math. 5 (1974), 49–55.Search in Google Scholar

Received: 2020-01-26
Accepted: 2020-05-09
Published Online: 2021-04-14
Published in Print: 2021-04-27

© 2021 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Regular papers
  2. Prof. RNDr. Michal Fečkan, DrSc. – Sexagenarian?
  3. Tribonacci numbers with two blocks of repdigits
  4. Padovan numbers that are concatenations of two distinct repdigits
  5. On the 2-rank and 4-rank of the class group of some real pure quartic number fields
  6. A general inverse matrix series relation and associated polynomials – II
  7. Some hardy type inequalities with finsler norms
  8. Starlikeness and convexity of the product of certain multivalent functions with higher-order derivatives
  9. Block Hessenberg matrices and spectral transformations for matrix orthogonal polynomials on the unit circle
  10. How is the period of a simple pendulum growing with increasing amplitude?
  11. Fourier transforms of convolution operators on orlicz spaces
  12. Some characterizations of property of trans-Sasakian 3-manifolds
  13. P-Adic metric preserving functions and their analogues
  14. On statistical convergence of sequences of closed sets in metric spaces
  15. A characterization of the uniform convergence points set of some convergent sequence of functions
  16. A nonparametric estimation of the conditional ageing intensity function in censored data: A local linear approach
  17. Donsker’s fuzzy invariance principle under the Lindeberg condition
  18. Characterization of generalized Gamma-Lindley distribution using truncated moments of order statistics
  19. Matrix variate pareto distributions
  20. Global exponential periodicity and stability of neural network models with generalized piecewise constant delay
  21. Optimal inequalities for contact CR-submanifolds in almost contact metric manifolds
Downloaded on 15.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2017-0468/html
Scroll to top button