250030 VO Algebra (2012W)

> The course continues and deepens the investigation of the basic algebraic structures groups, rings, fields. Emphasis will be placed on the investigation of fields an in particular of extensions of fields. Our aim is the main theorem of Galois theory which yields a description of a field extension via the corresponding symmetry group. Historically, the study of field extensions is motivated by the problem of solvability of algebraic equations by radicals and we want to describe the application of Galois theory to this problem. In particular, we will see that equations of degree 5 or bigger in general cannot be solved by radicals. Another topic will be group actions and the applications to the structure theory of finite groups ("Theorems of Sylow")