Primitive free cubics with specified norm and trace
HTML articles powered by AMS MathViewer
- by Sophie Huczynska and Stephen D. Cohen PDF
- Trans. Amer. Math. Soc. 355 (2003), 3099-3116 Request permission
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.References
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- 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
- 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, DOI 10.1007/s002000050112
- Stephen D. Cohen and Dirk Hachenberger, Primitivity, freeness, norm and trace, Discrete Math. 214 (2000), no. 1-3, 135–144. MR 1743632, DOI 10.1016/S0012-365X(99)00224-1
- S. D. Cohen and S. Huczynska, The primitive normal basis theorem — without a computer, J. London Math. Soc. 67 (2003), 41-56.
- S. D. Cohen and S. Huczynska, Primitive free quartics with specified norm and trace, Acta Arith. (to appear).
- H. Davenport, Bases for finite fields, J. London Math. Soc. 43 (1968), 21–39. MR 227144, DOI 10.1112/jlms/s1-43.1.21
- Nicholas M. Katz, Estimates for Soto-Andrade sums, J. Reine Angew. Math. 438 (1993), 143–161. MR 1215651, DOI 10.1515/crll.1993.438.143
- H. W. Lenstra Jr. and R. J. Schoof, Primitive normal bases for finite fields, Math. Comp. 48 (1987), no. 177, 217–231. MR 866111, DOI 10.1090/S0025-5718-1987-0866111-3
- 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
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
- MR Author ID: 50360
- Email: sdc@maths.gla.ac.uk
- Received by editor(s): September 26, 2002
- Received by editor(s) in revised form: January 30, 2003
- Published electronically: April 25, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 3099-3116
- MSC (2000): Primary 11T06; Secondary 11A25, 11T24, 11T30
- DOI: https://doi.org/10.1090/S0002-9947-03-03301-4
- MathSciNet review: 1974677