Class-group problems for cubic number fields

Class-number Problems for Cubic Number Fields

Class number approximation in cubic function fields

A central problem in number theory and algebraic geometry is the determination of the size of the group of rational points on the Jacobian of an algebraic curve over a finite field. This question also has applications to cryptography, since cryptographic systems based on algebraic curves generally require a Jacobian of non-smooth order in order to foil certain types of attacks. There a variety ...

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.

The 4-class Group of Real Quadratic Number Fields

In this paper we give an elementary proof of results on the structure of 4-class groups of real quadratic number fields originally due to A. Scholz. In a second (and independent) section we strengthen C. Maire’s result that the 2-class field tower of a real quadratic number field is infinite if its ideal class group has 4-rank ≥ 4, using a technique due to F. Hajir.

Computation of the Class Number and Class Group of a Complex Cubic Field

Let h and G be, respectively, the class number and the class group of a complex cubic field of discriminant A. A method is described which makes use of recent ideas of Lenstra and Schoof to develop fast algorithms for finding h and G. Under certain Riemann hypotheses it is shown that these algorithms will compute h in 0(|A|1/5 + E) elementary operations and G in 0(|A|l/4 + F) elementary operati...