Class Number and Ramification in Number Fields

Class-number Problems for Cubic Number Fields

Extensions of Number Fields with Wild Ramification of Bounded Depth

Fix a prime number p, a number field K, and a finite set S of primes of K. Let Sp be the set of all primes of K of residue characteristic p. Inside a fixed algebraic closure K of K, let KS be the maximal p-extension (Galois extension with pro-p Galois group) of K unramified outside S, and put GS = Gal(KS/K). The study of these “fundamental groups” is governed by a dichotomy between the tame (S∩...

Nilpotent Extensions of Number Fields with Bounded Ramification

We study a variant of the inverse problem of Galois theory and Abhyankar’s conjecture. If p is an odd rational prime and G is a finite p-group generated by s elements, s minimal, does there exist a normal extension L/Q such that Gal (L/Q) ∼= G with at most s rational primes that ramify in L? Given a nilpotent group of odd order G with s generators, we obtain a Galois extension L/Q with precisel...

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 ...

Ternary quadratic forms over number fields with small class number

We enumerate all positive definite ternary quadratic forms over number fields with class number at most 2. This is done by constructing all definite quaternion orders of type number at most 2 over number fields. Finally, we list all definite quaternion orders of ideal class number 1 or 2.