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Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Primitive free cubics with specified norm and trace

Author(s): Sophie Huczynska; Stephen D. Cohen
Journal: Trans. Amer. Math. Soc. 355 (2003), 3099-3116.
MSC (2000): Primary 11T06; Secondary 11A25, 11T24, 11T30
Posted: April 25, 2003
MathSciNet review: 1974677
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Abstract | References | Similar articles | Additional information

Abstract: The existence of a primitive free (normal) cubic $x^3-ax^2+cx-b$ over a finite field $F$ with arbitrary specified values of $a$ ($\neq 0$) and $b$ (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.

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L. Carlitz, Some problems involving primitive roots in a finite field, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 314-318, 618. MR 14:250f

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S. D. Cohen, Gauss sums and a sieve for generators of Galois fields, Publ. Math. Debrecen 56 (2000), 293-312. MR 2001e:11120

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S. D. Cohen and D. Hachenberger, Primitive normal bases with prescribed trace, Appl. Algebra Engrg. Comm. Comp. 9 (1999), 383-403. MR 2000c:11198

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S. D. Cohen and D. Hachenberger, Primitivity, freeness, norm and trace, Discrete Math. 214 (2000), 135-144. MR 2000j:11190

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S. D. Cohen and S. Huczynska, The primitive normal basis theorem -- without a computer, J. London Math. Soc. 67 (2003), 41-56.

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S. D. Cohen and S. Huczynska, Primitive free quartics with specified norm and trace, Acta Arith. (to appear).

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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: 10.1090/S0002-9947-03-03301-4
PII: S 0002-9947(03)03301-4
Received by editor(s): September 26, 2002
Received by editor(s) in revised form: January 30, 2003
Posted: April 25, 2003
Copyright of article: Copyright 2003, American Mathematical Society




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