Minimal Heights in Number Fields
Minimal heights of bases for number elds K over Q have been studied by Roy and Thunder, Masser, and Silverman, among others. The main results of this work focus on minimal heights of generator elements for number elds in general and quadratic number elds in particular.
It is shown that the minimal height of a generator of an imaginary quadratic extension Q(pd) of Q coincides with the minimal polynomial-height of the set of quadratic polynomials whose discriminant has squarefree-part (core) equal to d. This leads to a limit result concerning the size of the height of such a generator, using results of Ruppert. Invoking the Generalized Riemann Hypothesis in order to use an eective version of the Chebotarev Density Theorem allows for results of a non-limiting nature as well as a proof of a conjecture of Ruppert.
In addition, the heights of algebraic integers in quadratic extensions are analyzed and it is shown that the usual dichotomy between real and imaginary quadratic extensions exists here while this is no longer present when one considers general quadratic algebraic numbers.
A corollary of this is a characterization of which quadratic elds have algebraic integers
achieving the minimal height of a generator. A further corollary is a basic characterization, in terms of heights, of which imaginary quadratic elds have class number 1, contingent upon the Generalized Riemann Hypothesis.