Galoisgruppen von Eisensteinpolynomen über p-adischen Körpern

Galois theory has been a fundamental part of algebra and algebraic number theory for nearly 200 years since Évariste Galois. Nevertheless, the concrete calculation of the Galois group of a polynomial remains a challenging algorithmic problem. Significant progress has been made in the last ten years for polynomials over the rational numbers. Efficient methods for calculating Galois groups have now been implemented in common computer algebra systems. In contrast, no general methods for determining the Galois group are known for polynomials over p-adic fields. This book develops such computational methods for the first time and subjects them to a complexity analysis. Based on a comprehensive theoretical investigation of the structure of p-adic field extensions, the author presents algorithms for calculating the Galois group of Eisenstein polynomials. Numerous explicit examples of groups calculated using the new methods complete the content.