About this Course
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Advanced Level

Approx. 45 hours to complete

Suggested: 9 weeks of study, 4-8 hours/week...

English

Subtitles: English

100% online

Start instantly and learn at your own schedule.

Flexible deadlines

Reset deadlines in accordance to your schedule.

Advanced Level

Approx. 45 hours to complete

Suggested: 9 weeks of study, 4-8 hours/week...

English

Subtitles: English

Syllabus - What you will learn from this course

Week
1
1 hour to complete

Introduction

This is just a two-minutes advertisement and a short reference list.

...
1 video (Total 3 min), 2 readings
2 readings
Introduction/Manual10m
References10m
2 hours to complete

Week 1

We introduce the basic notions such as a field extension, algebraic element, minimal polynomial, finite extension, and study their very basic properties such as the multiplicativity of degree in towers.

...
6 videos (Total 84 min), 1 quiz
6 videos
1.4 Finite extensions. Algebraicity and finiteness.14m
1.5 Algebraicity in towers. An example.14m
1.6. A digression: Gauss lemma, Eisenstein criterion.13m
1 practice exercise
Quiz 140m
Week
2
2 hours to complete

Week 2

We introduce the notion of a stem field and a splitting field (of a polynomial). Using Zorn's lemma, we construct the algebraic closure of a field and deduce its unicity (up to an isomorphism) from the theorem on extension of homomorphisms.

...
5 videos (Total 67 min), 1 quiz
5 videos
2.4 Algebraic closure (continued).15m
2.5 Extension of homomorphisms. Uniqueness of algebraic closure.11m
1 practice exercise
QUIZ 240m
Week
3
4 hours to complete

Week 3

We recall the construction and basic properties of finite fields. We prove that the multiplicative group of a finite field is cyclic, and that the automorphism group of a finite field is cyclic generated by the Frobenius map. We introduce the notions of separable (resp. purely inseparable) elements, extensions, degree. We briefly discuss perfect fields. This week, the first ungraded assignment (in order to practice the subject a little bit) is given.

...
6 videos (Total 82 min), 1 reading, 1 quiz
6 videos
3.4 Separable elements.15m
3.5. Separable degree, separable extensions.15m
3.6 Perfect fields.9m
1 reading
Ungraded assignment 12h
1 practice exercise
QUIZ 340m
Week
4
2 hours to complete

Week 4

This is a digression on commutative algebra. We introduce and study the notion of tensor product of modules over a ring. We prove a structure theorem for finite algebras over a field (a version of the well-known "Chinese remainder theorem").

...
6 videos (Total 91 min), 1 quiz
6 videos
4.4 Examples. Tensor product of algebras.15m
4.5 Relatively prime ideals. Chinese remainder theorem.14m
4.6 Structure of finite algebras over a field. Examples.16m
1 practice exercise
QUIZ 440m
4.3
27 ReviewsChevron Right

Top reviews from Introduction to Galois Theory

By CLJun 16th 2016

Outstanding course so far - a great refresher for me on Galois theory. It's nice to see more advanced mathematics classes on Coursera.

Instructor

Avatar

Ekaterina Amerik

Professor
Department of Mathematics

About National Research University Higher School of Economics

National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communicamathematics, engineering, and more. Learn more on www.hse.ru...

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