Congruence of Ankeny-Artin-Chowla type modulo p^2



Abstract

In this paper the results of [4] and [5] on the congruence of Ankeny-Artin-Chowla type modulo p2 for real subfields of the Qp) of a prime degree l > 2 are simplified to explicit forms (2) and (3) of the Theorem 1. The congruence is then used to calculations of class numbers in special quintic fields and to some calculations in cubic fields.


Keywords

class number; subfields of cyclotomic fields

1. J.P. Buhler, R.E. Crandall, R.W. Sompolski, Irregular primes to one million, Math. Comp., 59 (1992), 717-722.
2. S. Jakubec, On divisibility of class number of real abelian fields of prime conductor, Abh. Math. Sem. Univ. Hamburg, 63 (1993), 67-86.
3. S. Jakubec, Congruence of Ankeny-Artin-Chowla type for cyclic fields of prime degree l, Math. Proc. Camb. Phil. Soc., 119 (1996), 17.
4. S. Jakubec, Congruence of Ankeny-Artin-Chowla type modulo p^2 for cyclic fields of prime degree l, Acta Arithmetica, LXXIV.4 (1996), 293-310.
5. S. Jakubec, Note on the congruence of Ankeny-Artin-Chowla type modulo p^2, Acta Arithmetica, LXXXV.4 (1998), 377-388.
6. E. Lehmer, Connection between Gaussian periods and cyclic units, Math. Comp., 50 (1988), 535-541.
7. W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 2nd edition, Springer-Verlag (1990).
8. R. Schoof, L.C. Washington, Quintic polynomials and real cyclotomic fields with large class number, Math. Comp., 50 (1988), 543-556.
9. L.C. Washington, Introduction to Cyclotomic Fields, 2nd edition, Springer-Verlag (1997).
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Published : 1998-09-30


JakubecS., & LaššákM. (1998). Congruence of Ankeny-Artin-Chowla type modulo p^2. Annales Mathematicae Silesianae, 12, 75-91. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14165

Stanislav Jakubec  jakubec@savba.savba.sk
Mathematical Institute, Slovak Academy of Sciences  Slovakia
Miroslav Laššák 
Mathematical Institute, Slovak Academy of Sciences  Slovakia



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