Introduction to Galois Theory II

We will examine essential ideas from rings and fields, such as polynomial rings, field extensions, and splitting fields, in this introductory course on Galois theory. Then, before defining Galois extensions, we’ll learn about normal and separable extensions. We’ll see several Galois group and Galois extension instances and structures.

The fundamental theorem of Galois theory, which establishes a relationship between Galois group subgroups and Galois extension intermediate fields, will then be proved.

After that, we’ll go over several key applications of Galois theory, such as quintic insolvability, Kummer extensions, and cyclotomic extensions. This course will place a strong emphasis on problem-solving exercises and providing numerous examples. We’ll give them various tasks to do, and we’ll have weekly problem-solving meetings where we’ll go over each topic in detail.

LAYOUT OF THE COURSE
Week 1: Rings and Fields Review I: polynomial rings, irreducibility criteria, algebraic elements, and field extensions I: polynomial rings, irreducibility criteria, algebraic elements, and field extensions

Week 2: Rings and Fields Review II: finite fields and field splitting

Week 3:Separable extensions vs. normal extensions

Week 4: Galois groups and fixed fields

Week 5: Galois extensions, properties, and examples

Week 6: Galois fundamental theorem

Week 7: Galois fundamental theorem

Week 8: Galois fundamental theorem

Week 9: Galois fundamental theorem