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Introduction to Galois Theory I complete course

  • Course level: All Levels

Description

Introduction to Galois Theory I

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

26 Lessons

Introduction to Galois Theory

mod01lec01 – Motivation and overview of the course00:00:00
mod01lec02 – Review of group theory00:00:00
mod01lec03 – Review of ring theory I00:00:00
mod01lec04 – Review of ring theory II00:00:00
mod01lec05 – Review of field theory I00:00:00
mod01lec06 – Review of field theory II00:00:00
mod01lec07 – Review of field theory III00:00:00
mod02lec08 – Problem Session 100:00:00
mod02lec09 – Problem Session 200:00:00
mod02lec10 – Beginning of Galois theory00:00:00
mod02lec11 – Fixed fields00:00:00
mod02lec12 – Theorem I on fixed fields00:00:00
mod02lec13 – Theorem II on fixed fields00:00:00
mod03lec14 – Galois extensions, Galois groups00:00:00
mod03lec15 – Normal extensions00:00:00
mod03lec16 – Problem Session – Part 300:00:00
mod03lec17 – Problem Session – Part 400:00:00
mod03lec18 – Separable extension – Part 100:00:00
mod03lec19 – Separable extension – Part 200:00:00
mod04lec20 – Characterization of Galois extensions – Part 100:00:00
mod04lec21 – Characterization of Galois extensions – Part 200:00:00
mod04lec22 – Examples of Galois extensions00:00:00
mod04lec23 – Motivating the main theorem of Galois theory00:00:00
mod04lec24 – Main theorem of Galois theory – Part 100:00:00
mod04lec25 – Main theorem of Galois theory – Part 200:00:00
mod04lec26 – Fundamental theorem of algebra00:00:00
Free

Enrolment validity: Lifetime