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

Full-text PDF Free Access

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.***Algebraic number theory*, Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union. Edited by J. W. S. Cassels and A. Fröhlich, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR**0215665****2.**John L. Kelley and T. P. Srinivasan,*Measure and integral. Vol. 1*, Graduate Texts in Mathematics, vol. 116, Springer-Verlag, New York, 1988. MR**918770****3.**Sergey Konyagin and Igor Shparlinski,*On the Distribution of Residues of Finitely Generated Multiplicative Groups and Some of Their Applications*, to appear.**4.**Serge Lang,*Algebraic number theory*, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR**1282723****5.**C. G. Lekkerkerker,*Geometry of numbers*, Bibliotheca Mathematica, Vol. VIII, Wolters-Noordhoff Publishing, Groningen; North-Holland Publishing Co., Amsterdam-London, 1969. MR**0271032****6.**Raphael M. Robinson,*Numbers having 𝑚 small 𝑚th roots mod 𝑝*, Math. Comp.**61**(1993), no. 203, 393–413. MR**1189522**, https://doi.org/10.1090/S0025-5718-1993-1189522-0**7.**P. Stevenhagen and H.W. Lenstra, Jr.,*Chebotarev and his density theorem*, Math. Intelligencer**18**(1996), 26-37. CMP**96:14****8.**Jeffrey D. Vaaler,*A geometric inequality with applications to linear forms*, Pacific J. Math.**83**(1979), no. 2, 543–553. MR**557952**

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
11A07,
11A15,
11N05,
11R18,
11R44

Retrieve articles in all journals with MSC (1991): 11A07, 11A15, 11N05, 11R18, 11R44

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