On certain character sums over

Author:
Chih-Nung Hsu

Journal:
Proc. Amer. Math. Soc. **126** (1998), 647-652

MSC (1991):
Primary 11A07; Secondary 11L40, 11N05

DOI:
https://doi.org/10.1090/S0002-9939-98-04582-1

MathSciNet review:
1469411

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the finite field with elements and let denote the ring of polynomials in one variable with coefficients in . Let be a monic polynomial irreducible in . We obtain a bound for the least degree of a monic polynomial irreducible in ( odd) which is a quadratic non-residue modulo . We also find a bound for the least degree of a monic polynomial irreducible in which is a primitive root modulo .

**1.**N. C., Ankeny `The Least Quadratic Non Residue', Annals of Mathematics, Vol 55, No. 1 (1952), pp. 65-72. MR**13:538c****2.**E. Artin, `Quadratische Körper im Gebiete der höheren Kongruenzen I, II', Math. Zeitschrift 19 (1924), pp. 153-246.**3.**G. W. Effinger and D. R. Hayes, `Additive Number Theory of Polynomials Over a Finite Field', Oxford, Clarendon Press (1991). MR**92k:11103****4.**G. H. Hardy and E. M. Wright, `An Introduction to the Theory of Numbers', Oxford, Clarendon Press (1945). MR**16:673c (3rd ed.)****5.**S. A. Stepanov, `Arithmetic Of Algebraic Curves', Translated from Russian by Irene Aleksanova, Plenum Publishing Corporation (1994). MR**95j:11055**

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Additional Information

**Chih-Nung Hsu**

Affiliation:
Department of Mathematics, National Taiwan Normal University, 88 Sec. 4 Ting-Chou Road, Taipei, Taiwan

Email:
maco@math.ntnu.edu.tw

DOI:
https://doi.org/10.1090/S0002-9939-98-04582-1

Keywords:
Riemann Hypothesis,
quadratic non-residues,
primitive roots

Received by editor(s):
August 20, 1996

Communicated by:
Dennis A. Hejhal

Article copyright:
© Copyright 1998
American Mathematical Society