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

Admissible Primes and Euclidean Quadratic Fields


Affiliations
1 Department of Mthematics and Statistics, Jeffery Hall, Queen’s University, Kingston, Ontario, K7L 3N6, Canada
2 Institute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai - 600113, India
3 Chennai Mathematical Institute, SIPCOT IT Park, Siruseri, Chennai - 603103, India
     

   Subscribe/Renew Journal


Let K be a real quadratic field with ring of integers ΟK . We exhibit an infinite family of real quadratic fields K, such that ΟK contains an admissible set of primes with two elements.We then study the implications of this construction to the determination of Euclidean real quadratic fields and related questions.
User
Subscription Login to verify subscription
Notifications
Font Size

  • V. Buniakovsky, Sur les diviseurs numeriques invariables des fonctions rationnelles entieres, Mem Acad. Sci. St Petersburg, 6 (1857) 305–329.

  • David A. Clark and M. Ram Murty, The Euclidean algorithm for Galois extensions, Journal fur die reine und angewandte Mathematik, 459 (1995) 151–162.

  • David S. Dummit and Richard M. Foote, Abstract Algebra, John Wiley & Sons (2004).

  • Jody Esmonde and M. Ram Murty, Problems in Algebraic Number Theory, Graduate texts in Mathematics, Springer Science and Business Media (2005).

  • Rajiv Gupta, M. Ram Murty and V. Kumar Murty, The Euclidean algorithm for S-integers In: Number Theory (Montreal, June 1985), CMS Conf. Proc. 7, Amer. Math. Soc. (1987) 189–201.

  • Malcom Harper, Z[√14] is Euclidean, Canad. J. Math., Vol. 56 (2004) no. 1, 55–70.

  • M. Harper and M. Ram Murty Euclidean rings of algebraic integers, Canadian Journal of Math., 56 (2004) no. 1, 71–76.

  • C. Hooley, On Artin’s conjecture, J. Reine Agew. Math., 225 (1967) 209–220.

  • Franz Lemmermeyer, The euclidean algorithm in algebraic number fields, Exposition. Math., 13 (1995) no. 5, 385–416

  • Daniel A. Marcus, Number Fields, Graduate texts in mathematics, Springer-Verlag (1977).

  • T. Motzkin, The Euclidean algorithm, Bull. Am. Math. Soc., 55 (1949) no. 12, 1142–1146.

  • M. RamMurty and V. Kumar Murty, A variant of the Bombieri-Vinogradov theorem, in Number Theory, Proceedings of the 1985 Montreal Conference, 7 (1987) 243–272.

  • M. Ram Murty and Kathleen Petersen, A Bombieri-Vinogradov theorem for all number fields, Transactions of the American Math. Society, 365 (2013) no. 9, 4987–5032.

  • T. Nagell, Zur Arithmetik der Polynome, Abhandl. Math. Sem. Hamburg, 1 (1922) 179–194.

  • G. Ricci, Ricerche aritmetiche sui polinomi, Rend. Circ. Mat. Palermo, 57 (1933) 433–475.

  • Pierre Samuel, About Euclidean rings, Journal of Algebra, Vol. 19 Issue 2 (1971) 282–301.

  • P. J. Weinberger, On Euclidean rings of algebraic integers, Proc. Symp. Pure Math., Analytic number theory, AMS, 24 (1973) 321–332 6.

  • Leonardo Zapponi, Parametric solutions of Pell equations arXiv:1503.00637v1.


Abstract Views: 318

PDF Views: 0




  • Admissible Primes and Euclidean Quadratic Fields

Abstract Views: 318  |  PDF Views: 0

Authors

M. Ram Murty
Department of Mthematics and Statistics, Jeffery Hall, Queen’s University, Kingston, Ontario, K7L 3N6, Canada
Kotyada Srinivas
Institute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai - 600113, India
Muthukrishnan Subramani
Chennai Mathematical Institute, SIPCOT IT Park, Siruseri, Chennai - 603103, India

Abstract


Let K be a real quadratic field with ring of integers ΟK . We exhibit an infinite family of real quadratic fields K, such that ΟK contains an admissible set of primes with two elements.We then study the implications of this construction to the determination of Euclidean real quadratic fields and related questions.

References