Certain pure cubic fields with class-number one

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, ...

Binomial squares in pure cubic number fields

Let K = Q(ω), with ω3 = m a positive integer, be a pure cubic number field. We show that the elements α ∈ K× whose squares have the form a − ω for rational numbers a form a group isomorphic to the group of rational points on the elliptic curve Em : y2 = x3 − m. This result will allow us to construct unramified quadratic extensions of pure cubic number fields K.

Class-number Problems for Cubic Number Fields

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.

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 ...