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Bounds for Multiplicative Cosets over Fields of Prime Order

Author: Corey Powell
Journal: Math. Comp. 66 (1997), 807-822
MSC (1991): Primary 11A07, 11A15; Secondary 11N05, 11R18, 11R44
MathSciNet review: 1372008
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Abstract: Let $m$ be a positive integer and suppose that $p$ is an odd prime with $p \equiv 1 \bmod m$. Suppose that $a \in (\Bbb Z /p\Bbb Z )^{\ast } $ and consider the polynomial $x^m-a$. If this polynomial has any roots in $(\Bbb Z /p\Bbb Z )^{\ast } $, where the coset representatives for $\Bbb Z /p\Bbb Z $ are taken to be all integers $u$ with $|u|<p/2$, then these roots will form a coset of the multiplicative subgroup $\mu _m$ of $(\Bbb Z /p\Bbb Z )^{\ast } $ consisting of the $m$th roots of unity mod $p$. Let $C$ be a coset of $\mu _m$ in $(\Bbb Z /p\Bbb Z )^{\ast } $, and define $|C|=\max _{u \in C}{|u|}$. In the paper ``Numbers Having $m$ Small $m$th Roots mod $p$'' (Mathematics of Computation, Vol. 61, No. 203 (1993),pp. 393-413), Robinson gives upper bounds for $M_1(m,p)=\min _{\tiny C \in (\Bbb Z /p\Bbb Z )^{\ast } /\mu _m }{|C|}$ of the form $M_1(m,p)<K_mp^{1-1/\phi (m)}$, where $\phi $ is the Euler phi-function. 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 $m$ is a power of $2$ or when $m=6.$

References [Enhancements On Off] (What's this?)

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Additional Information

Corey Powell
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720

Received by editor(s): May 30, 1995
Received by editor(s) in revised form: January 26, 1996
Article copyright: © Copyright 1997 American Mathematical Society