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

DOI:
https://doi.org/10.1090/S0025-5718-97-00797-7

MathSciNet review:
1372008

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Abstract | References | Similar Articles | Additional Information

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. 393-413), Robinson gives upper bounds for of the form , where 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 is a power of or when

**1.**J.W.S. Cassels and A. Frohlich,*Algebraic Number Theory*, Academic Press Limited, 1967. MR**35:6500****2.**John L. Kelley and T.P. Srinivasan,*Measure and Integral*, Springer-Verlag, 1988. MR**89e:28001****3.**Sergey Konyagin and Igor Shparlinski,*On the Distribution of Residues of Finitely Generated Multiplicative Groups and Some of Their Applications*, to appear.**4.**S. Lang,*Algebraic Number Theory*, Springer-Verlag, 1994. MR**95f:11085****5.**C.G. Lekkerkerker,*Geometry of Numbers*, Wolters-Noordhoff and North-Holland Publishing Companies, 1969. MR**42:5915****6.**R.M. Robinson,*Numbers Having Small th Roots mod*, Mathematics of Computation**61**(1993), no. 203, 393-413. MR**93k:11002****7.**P. Stevenhagen and H.W. Lenstra, Jr.,*Chebotarev and his density theorem*, Math. Intelligencer**18**(1996), 26-37. CMP**96:14****8.**J.E. Vaaler,*A Geometric Inequality with Applications to Linear Forms*, Pacific Journal of Mathematics**83**(1979), no. 2, 543-553. MR**81d:52007**

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Additional Information

**Corey Powell**

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

DOI:
https://doi.org/10.1090/S0025-5718-97-00797-7

Received by editor(s):
May 30, 1995

Received by editor(s) in revised form:
January 26, 1996

Article copyright:
© Copyright 1997
American Mathematical Society