Ing. Magdaléna Tinková, Ph.D.

2022 - 2025

The goal of this project is to study number fields of higher degrees and use the local-global principle to determine their structure. We will focus on composite fields, for which one can extend partial information from their subfields to the whole field. For them, we will investigate their additively indecomposable integers and derive asymptotic formulas for the number of elements of small norms. Moreover, we plan to estimate their Pythagoras number and the number of variables of their universal quadratic forms. This was studied by many great mathematicians, including Lagrange or Siegel, but we still do not understand it fully. The methodology is based, for example, on the Hasse norm principle or representation of quadratic forms. Furthermore, considering a system of linear polynomials in one variable, we will aim to find an asymptotic formula for the number of integers up to some bound, for which the values of these polynomials are primes with given prescribed primitive root. For that, we want to use the nilpotent circle method developed by Green, Tao, and Ziegler.