Bounds for Multiplicative Cosets over Fields of Prime Order
Author:
Corey Powell
Journal:
Math. Comp. 66 (1997), 807822
MSC (1991):
Primary 11A07, 11A15; Secondary 11N05, 11R18, 11R44
MathSciNet review:
1372008
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Abstract: Let be a positive integer and suppose that is an odd prime with . Suppose that and consider the polynomial . If this polynomial has any roots in , where the coset representatives for are taken to be all integers with , then these roots will form a coset of the multiplicative subgroup of consisting of the th roots of unity mod . Let be a coset of in , and define . In the paper ``Numbers Having Small th Roots mod '' (Mathematics of Computation, Vol. 61, No. 203 (1993),pp. 393413), Robinson gives upper bounds for of the form , where is the Euler phifunction. This paper gives lower bounds that are of the same form, and seeks to sharpen the constants in the upper bounds of Robinson. The upper bounds of Robinson are proven to be optimal when is a power of or when
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Additional Information
Corey Powell
Affiliation:
Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
DOI:
http://dx.doi.org/10.1090/S0025571897007977
PII:
S 00255718(97)007977
Received by editor(s):
May 30, 1995
Received by editor(s) in revised form:
January 26, 1996
Article copyright:
© Copyright 1997 American Mathematical Society
