Computational Galois Theory for Local Fields

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Galois groups are fundamental mathematical objects, which provide information about the solvability of polynomials by radicals. The applicant has gained respectable progress in computing intermediate fields and Galois groups over rational numbers in the past few years. While the recent implementations provides computations of Galois groups for polynomials up to high double-digit degree, these computations are difficult to perform efficiently, and it is unknown if these algorithms can have polynomial time complexity. For the computation of intermediate fields, the applicant developed a new algorithm, which de-livers a system of generating subfields in polynomial complexity. Then any arbitrary subfield can be described as a suitable intersection of the generating subfields. Furthermore, the running time for computing all subfields is proportional to the number of subfields. For this project at hand, we aim to develop and implement nontrivial algorithms for the computation of subfields and Galois groups over local fields, i.e. over p-adic fields and local function fields. It is fair to hope that these algorithms provide improvements and better understanding for the respective algorithms over global fields. This project requires a very detailed investigation of the local fields' structure, and a fine intuition for retrieving effectiveness for computation over local fields. This yields a nice interaction of theory and computer algebra applications. The computation of intermediate fields leads back to factorisation of polynomials and solutions of linear systems of equations. Hereby, recent implementations of factorisation algorithms over local fields only provide approximations, and as these are the input of the linear system of equations, we have to consider precision problems like in numerical analysis. The main problem for the computation of Galois groups over local fields is that we do not have easy access to the zeros and their approximations, which does not allow the transformation of known algorithms for global fields. We would like to attack the computation of Galois group via the knowledge of the absolute Galois group and local class field theory. The applicant administrates a database for number fields filled with over 2 million polynomials. This database shall be extended and be expanded by local function fields with small characteristic. These data are very important to find and understand interesting examples for conjectures within geometry and number theory. Furthermore, big tables are useful to make and test conjectures about the asymptotics of such objects.