Class number approximation in cubic function fields

Kummer, Eisenstein, Computing Gauss Sums as Lagrange Resolvents

In fact, [Eisenstein 1850] evaluated cubes and fourth powers of Gauss sums attached to cubic and quartic characters to prove the corresponding reciprocity laws. One essential point is the p-adic approximation of Gauss sums by [Kummer 1847], generalized in [Stickelberger 1890]. Since the rings of algebraic integers generated by third or fourth roots of unity have class number one and finitely-ma...

A Lower Bound for the Class Number of Certain Cubic Number Fields

Let AT be a cyclic number field with generating polynomial i a— 3 ^ û + 3 x3 —Y-x1 -=~-xi and conductor m. We will derive a lower bound for the class number of these fields and list all such fields with prime conductor m = (a1 + 21)/A or m = (1 + 21b2)/A and small class number.

Number fields with large class groups

After a review of the quadratic case, a general problem about the existence of number fields of a fixed degree with extremely large class numbers is formulated. This problem is solved for abelian cubic fields. Then some conditional results proven elsewhere are discussed about totally real number fields of a fixed degree, each of whose normal closure has the symmetric group as Galois group.

Upper bounds for residues of Dedekind zeta functions and class numbers of cubic and quartic number fields

Let K be an algebraic number field. Assume that ζK(s)/ζ(s) is entire. We give an explicit upper bound for the residue at s = 1 of the Dedekind zeta function ζK(s) of K. We deduce explicit upper bounds on class numbers of cubic and quartic number fields.

Class-number Problems for Cubic Number Fields

Research Statement and Plan

My main research interest is number theory, in particular algebraic and computational number theory. Specifically, I am interested in computational aspects of number fields and function fields, in particular field tabulation and efficient computation of invariants associated with number fields and function fields. Many problems in this area have been explored extensively in the case of number f...