Journal of the Ramanujan Mathematical Society

Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

Distribution of Prime Ideals Across Ideal Classes in the Class Groups


Affiliations
1 Department of Mathematics, IISER Berhampur, Engineering School Road, Khodasingi, Brahmapur 760 010, India
     

   Subscribe/Renew Journal


In this article we investigate the distribution of prime ideals of residue degree bigger than one across the ideal classes in the class group of a number field L. A criterion for the class group of L being generated by the classes of prime ideals of residue degree f > 1 is provided. Further, some consequences of this study on the solvability of norm equations for L/ℚ and on the problem of finding annihilators for relative extensions are discussed.
User
Subscription Login to verify subscription
Notifications
Font Size

  • V. Acciaro, Solvability of norm equations over cyclic number fields of prime degree, Mathematics of Computation, 65 no. 216, (1996) 1663–1674.

  • Y. Bilu, Catalan’s conjecture (after Mihailescu), Seminare Bourbaki (2002–2003).

  • T. D. Browning and R. Newton, The proportion of failures of the Hasse norm principle, Mathematika, 62 no. 2, (2016) 337–347.

  • S. Dasgupta, Stark’s conjectures, B. A. (with Honors) thesis, Harvard university (1999).

  • C. Fieker, A. Jurk and M. Pohst, On solving relative norm equations in algebraic number fields, Mathematics of Computation, 66 no. 217, (1997) 399–410.

  • D. A. Garbanati, An algorithm for finding an algebraic number whose norm is a given rational number, J. Reine Angew. Math., 316 (1980) 1–13.

  • G. J. Janusz, Algebraic number fields, second edition, Graduate Studies in Mathematics, American Mathematical Society, Vol. 7 (1996),

  • E. E. Kummer, Collected papers, Vol. I. Springer Verlag, Berlin 1975.

  • S. Lang, Cyclotomic fields 1 and 2, 2nd ed., Grad. Texts in Math., Springer, New York, 123 (1990).

  • S. Lang, Algebraic number theory, 2nd ed., Grad. Texts in Math., Springer, New York, 110 (1994).

  • H. W. Lenstra Jr. and P. Stevenhagen, Primes of degree one and algebraic cases of Chebotarev’s Theorem, L’ Enseignement Mathematique, t., 37 (1991) 17–30.

  • J. C. Miller, Class numbers of real cyclotomic fields of composite conductor, LMS J. Comput. Math., suppl. A, 17 (2014) 404–417.

  • M. Rosen, Class groups in cyclic 𝓁-extensions: comments on a paper by G. Cornell, Proc. Amer. Math. Soc., 142 no. 1, (2014) 21–18.

  • J. W. Sands, Galois groups of exponent two and the Brumer-Stark conjecture, J. Reine Angew. Math., 349 (1984) 129–135.

  • R. Schoof, Catalan’s conjecture, Universitext, Springer (2008).

  • John Tate, Brumer-Stark-Stickelberger, Seminaire de Theorie des Nombres de Bordeaux, 10 (1980–1981) 1–16.

  • L. C. Washington, Introduction to Cyclotomic fields, Second Edition, Springer (1991).


Abstract Views: 258

PDF Views: 0




  • Distribution of Prime Ideals Across Ideal Classes in the Class Groups

Abstract Views: 258  |  PDF Views: 0

Authors

Prem Prakash Pandey
Department of Mathematics, IISER Berhampur, Engineering School Road, Khodasingi, Brahmapur 760 010, India

Abstract


In this article we investigate the distribution of prime ideals of residue degree bigger than one across the ideal classes in the class group of a number field L. A criterion for the class group of L being generated by the classes of prime ideals of residue degree f > 1 is provided. Further, some consequences of this study on the solvability of norm equations for L/ℚ and on the problem of finding annihilators for relative extensions are discussed.

References