Abstract
For any square-free polynomial D over a finite field of characteristic at least 5, we present an algorithm for generating all cubic function fields of discriminant D. We also provide a count of all these fields according to their splitting at infinity. When D′ = D/(-3) has even degree and a leading coefficient that is a square, i.e. D′ is the discriminant of a real quadratic function field, this method makes use of the infrastructures of this field. This infrastructure method was first proposed by Shanks for cubic number fields in an unpublished manuscript from the late 1980s. While the mathematical ingredients of our construction are largely classical, our algorithm has the major computational advantage of finding very small minimal polynomials for the fields in question.
| Original language | English |
|---|---|
| Pages (from-to) | 1839-1885 |
| Number of pages | 47 |
| Journal | International Journal of Number Theory |
| Volume | 11 |
| Issue number | 6 |
| DOIs | |
| State | Published - 27 Sep 2015 |
Bibliographical note
Funding Information:This work was inspired by, and is dedicated to, the late Dan Shanks. The first, third and fourth authors are supported by NSERC of Canada. The second author is the corresponding author and supported by the LG Yonam Foundation. The authors thank Pieter Rozenhart for discussions and some test computations that aided in our algorithm comparisons of Sec. 6.3. We are also grateful to an anonymous referee for his or her careful review and constructive suggestions for improvement of our paper.
Publisher Copyright:
© 2015 World Scientific Publishing Company.
Keywords
- Cubic function field
- discriminant
- quadratic function field
- quadratic generator
- reduced ideal
- signature