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A Note on Bernoulli-Goss Polynomials

Published online by Cambridge University Press:  20 November 2018

K. Ireland  and
D. Small
K. Ireland
Affiliation:
Department of Mathematics and Statistics, University of New BrunswickP. O. Box 4400, FrederictonNew Brunswick E3B 5 A3
D. Small
Affiliation:
Department of Mathematics and Statistics, University of New BrunswickP. O. Box 4400, FrederictonNew Brunswick E3B 5 A3
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Abstract

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In an important series of papers ([3], [4], [5]), (see also Rosen and Galovich [1], [2]), D. Goss has developed the arithmetic of cyclotomic function fields. In particular, he has introduced Bernoulli polynomials and proved a non-existence theorem for an analogue to Fermat’s equation for regular “exponent”. For each odd prime p and integer n, l ≤ n ≤ p2-2 we derive a closed form for the nth Bernoulli polynomial. Using this result a computer search for regular quadratic polynomials of the form x2-a was made. For primes less than or equal to 269 regular quadratics exist for p= 3, 5, 7, 13, 31.

Keywords

12C15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 2 , 01 June 1984 , pp. 179 - 184
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Galovich, S. and Rosen, M., The Class Number of Cyclotomic Function Fields, Journal of Number Theory, Vol. 13, No. 3, August 1981, 363–375.Google Scholar
2. Galovich, S. and Rosen, M., Unit and Class Groups in Cyclotomic Function Fields, Journal of Number Theory, Vol. 14, No. 2, 1982.Google Scholar
3. Goss, D., On a Fermät Equation Arising in the Arithmetic Theory of Function Fields, Math. Ann. 261, 269–286 (1982).Google Scholar
4. Goss, D., v-adic Zeta Functions, L-series and Measures for Function Fields, Inven. Math. 55, 107–116 (1979), pp. 107–119.Google Scholar
5. Goss, D., The Arithmetic of Function Fields 2: The Cyclotomic Theory, Journal of Algebra, 81, (1), 1983, 107–149.Google Scholar
6. Goss, D., On a New Type of L-function for Algebraic Curves Over Finite Fields, Pacific Journal of Math.Google Scholar
7. Thomas, E., On the Zeta Function for Function Fields Over , Pacific Journal of Math., 107, (1), 1983, 251–256.Google Scholar