Primitive free cubics with specified norm and trace

Authors:
Sophie Huczynska and Stephen D. Cohen

Journal:
Trans. Amer. Math. Soc. **355** (2003), 3099-3116

MSC (2000):
Primary 11T06; Secondary 11A25, 11T24, 11T30

Published electronically:
April 25, 2003

MathSciNet review:
1974677

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The existence of a primitive free (normal) cubic over a finite field with arbitrary specified values of () and (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.

**[Ca1]**L. Carlitz,*Primitive roots in a finite field*, Trans. Amer. Math. Soc.**73**(1952), 373–382. MR**0051869**, 10.1090/S0002-9947-1952-0051869-9**[Ca2]**L. Carlitz,*Some problems involving primitive roots in a finite field*, Proc. Nat. Acad. Sci. U.S.A.**38**(1952), 314–318; errata, 618. MR**0049939****[Co]**Stephen D. Cohen,*Gauss sums and a sieve for generators of Galois fields*, Publ. Math. Debrecen**56**(2000), no. 3-4, 293–312. Dedicated to Professor Kálmán Győry on the occasion of his 60th birthday. MR**1765983****[CoHa1]**S. D. Cohen and D. Hachenberger,*Primitive normal bases with prescribed trace*, Appl. Algebra Engrg. Comm. Comput.**9**(1999), no. 5, 383–403. MR**1697177**, 10.1007/s002000050112**[CoHa2]**Stephen D. Cohen and Dirk Hachenberger,*Primitivity, freeness, norm and trace*, Discrete Math.**214**(2000), no. 1-3, 135–144. MR**1743632**, 10.1016/S0012-365X(99)00224-1**[CoHu1]**S. D. Cohen and S. Huczynska,*The primitive normal basis theorem -- without a computer*, J. London Math. Soc.**67**(2003), 41-56.**[CoHu2]**S. D. Cohen and S. Huczynska,*Primitive free quartics with specified norm and trace*, Acta Arith. (to appear).**[Da]**H. Davenport,*Bases for finite fields*, J. London Math. Soc.**43**(1968), 21–39. MR**0227144****[Ka]**Nicholas M. Katz,*Estimates for Soto-Andrade sums*, J. Reine Angew. Math.**438**(1993), 143–161. MR**1215651**, 10.1515/crll.1993.438.143**[LeSc]**H. W. Lenstra Jr. and R. J. Schoof,*Primitive normal bases for finite fields*, Math. Comp.**48**(1987), no. 177, 217–231. MR**866111**, 10.1090/S0025-5718-1987-0866111-3**[LiNi]**Rudolf Lidl and Harald Niederreiter,*Finite fields*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 20, Cambridge University Press, Cambridge, 1997. With a foreword by P. M. Cohn. MR**1429394**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
11T06,
11A25,
11T24,
11T30

Retrieve articles in all journals with MSC (2000): 11T06, 11A25, 11T24, 11T30

Additional Information

**Sophie Huczynska**

Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland

Address at time of publication:
School of Informatics, University of Edinburgh, Edinburgh EH8 9LE, Scotland

Email:
shuczyns@inf.ed.ac.uk

**Stephen D. Cohen**

Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland

Email:
sdc@maths.gla.ac.uk

DOI:
https://doi.org/10.1090/S0002-9947-03-03301-4

Received by editor(s):
September 26, 2002

Received by editor(s) in revised form:
January 30, 2003

Published electronically:
April 25, 2003

Article copyright:
© Copyright 2003
American Mathematical Society