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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

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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
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