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SMALLEST IRREDUCIBLE OF THE FORM x²-dy²

SHANSHAN DING

Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA

https://doi.org/10.1142/S1793042109002158Cited by:0 (Source: Crossref)

Abstract

It is a classical result that prime numbers of the form x² + ny² can be characterized via class field theory for an infinite set of n. In this paper, we derive the function field analogue of the classical result. Then, we apply an effective version of the Chebotarev density theorem to bound the degree of the smallest irreducible of the form x² - dy², where x, y, and d are elements of a polynomial ring over a finite field.

Keywords:

AMSC: 11R37, 11R29

References

D. A. Cox , Primes of the Form x2 + ny2 ( Wiley , New York , 1989 ) . Google Scholar
J. C. Lagarias and A. M. Odlyzko, Algebraic Number Fields, ed. A. Frölich (Academic Press, London, 1977) pp. 409–464. Google Scholar
K. Murty and J. Scherk, C. R. Acad. Sci. Paris 319, 523 (1994). Web of Science, Google Scholar
M. Rosen, Expo. Math. 5, 365 (1987). Google Scholar
M. Rosen , Number Theory in Function Fields ( Springer-Verlag , New York , 2002 ) . Crossref, Google Scholar