The student should acquire a good knowledge about archimedean and non archimedean absolute values and valuations and about the completion of fields with respect to such absolute values. The students should be able to apply such knowledge to the investigation of the structure and main properties of complete fields with particular emphasis on the p-adic fields.
The student should acquire a good knowledge about Galois extensions, Galois groups and the fundamental theorem of Galois theory (in the infinite case as well). The student should be able to apply such knowledge to the investigation of various extensions (radical extensions, constructible, cyclic, abelian, cyclotomic,...).
After the lectures the student should be able to present the topics of the course with clarity and precision and with an appropriate specific scientific language and to improve his/her knowledge in local fields and Galois theory by consulting the existing literature on the subject.