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Introduction to Galois Theory II complete course

  • Course level: All Levels

Description

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

Topics for this course

22 Lessons

mod05lec27 – Problem Session – Part 5

mod05lec28 – Problem Session – Part 600:00:00
mod05lec29 – Problem Session – Part 700:00:00
mod05lec30 – Problem Session – Part 800:00:00
mod05lec31 – Problem Session – Part 900:00:00
mod05lec32 – Kummer extensions – Part 100:00:00
mod06lec33 – Kummer extensions – Part 200:00:00
mod06lec34 – Kummer extensions – Part 300:00:00
mod06lec35 – Cyclotomic extensions – Part 100:00:00
mod06lec36 – Cyclotomic extensions – Part 200:00:00
mod06lec37 – Solvability by radicals00:00:00
mod06lec38 – Characterizations of solvability – Part 100:00:00
mod07lec39 – Characterizations of solvability – Part 200:00:00
mod07lec40 – Discriminants, Galois groups of polynomials00:00:00
mod07lec41 – Quartics are solvable00:00:00
mod07lec42 – Solvable groups – Part 100:00:00
mod07lec43 – Solvable groups – Part 200:00:00
mod07lec44 – Solvable groups – Part 300:00:00
mod08lec45 – Insolvability of quintics00:00:00
mod08lec46 – Problem Session – Part 1000:00:00
mod08lec47 – Problem Session – Part 1100:00:00
mod08lec48 – Problem Session – Part 1200:00:00
mod08lec49 – Problem Session – Part 1300:00:00
Free

Enrolment validity: Lifetime