Algorithmische Algebra: Wintersemester 2018/19

Gruppen Operationen Definition Gruppe, symmetrische Gruppen Untergruppen, Erzeugung, Homomorphie Freie Gruppen Definition Operation, Bahnenalgorithmus Nebenklassen und der Satz von Hölder Stabilisatoren Klassifikation transitiver Operationen Erzeuger für Stabilisatoren Beispiel von Operationen und Untergruppen, Anwendungen Die Konjugationsoperation Anwendung auf kombinatorische Abzählprobleme Homomorphiesatz Stabilisatorketten Die Sylowschen Sätze Gröbner bases Monomials and monomial ideals Monomial orderings Division with remainder Gröbner bases and Buchberger's algorithm First applications: Submodule membership and syzygies Iterated syzygies and Hilbert's Syzygy Theorem Elimination Constructive Abelian categories Categories The category of modules Objects Morphisms Subobjects and factor objects Kernels and cokernels, images and co-images Products and coproducts Pull-backs and push-outs as kernels and cokernels More categorical notions Additive and Abelian categories Computable rings A constructively Abelian category of matrices Functors Natural transformations Exact sequences, homology, and cohomology Free resolutions