3-Class groups of cubic cyclic function fields

On 3-class Groups of Certain Pure Cubic Fields

Let p be a prime number, and let K = Q( 3 √p). Let M = Q(ζ, 3 √p) = Q( √ −3, 3 √p), where ζ is a primitive cube root of unity. Let SK be the 3-class group of K (that is, the Sylow 3-subgroup of the ideal class group of K). Let SM (respectively, SQ(ζ)) be the 3-class group ofM (respectively, Q(ζ)). Since Q(ζ) has class number 1, then SQ(ζ) = {1}. Assuming p ≡ 1 (mod 9), Calegari and Emerton [3, ...

Properties of Class Groups of a Family of Cyclic Cubic 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 ...

Quadratic Fields with Cyclic 2-class Groups

For any integer k ≥ 1, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order 2. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class to be a square. In memory of David Hayes.

Tame kernels of cubic cyclic fields

There are many results describing the structure of the tame kernels of algebraic number fields and relating them to the class numbers of appropriate fields. In the present paper we give some explicit results on tame kernels of cubic cyclic fields. Table 1 collects the results of computations of the structure of the tame kernel for all cubic fields with only one ramified prime p, 7 ≤ p < 5, 000....