About this course: A very beautiful classical theory on field extensions of a certain type (Galois extensions) initiated by Galois in the 19th century. Explains, in particular, why it is not possible to solve an equation of degree 5 or more in the same way as we solve quadratic or cubic equations. You will learn to compute Galois groups and (before that) study the properties of various field extensions. We first shall survey the basic notions and properties of field extensions: algebraic, transcendental, finite field extensions, degree of an extension, algebraic closure, decomposition field of a polynomial. Then we shall do a bit of commutative algebra (finite algebras over a field, base change via tensor product) and apply this to study the notion of separability in some detail. After that we shall discuss Galois extensions and Galois correspondence and give many examples (cyclotomic extensions, finite fields, Kummer extensions, Artin-Schreier extensions, etc.). We shall address the question of solvability of equations by radicals (Abel theorem). We shall also try to explain the relation to representations and to topological coverings. Finally, we shall briefly discuss extensions of rings (integral elemets, norms, traces, etc.) and explain how to use the reduction modulo primes to compute Galois groups. PREREQUISITES A first course in general algebra — groups, rings, fields, modules, ideals. Some knowledge of commutative algebra (prime and maximal ideals — first few pages of any book in commutative algebra) is welcome. For exercises we also shall need some elementary facts about groups and their actions on sets, groups of permutations and, marginally, the statement of Sylow's theorems. ASSESSMENTS A weekly test and two more serious exams in the middle and in the end of the course. For the final result, tests count approximately 30%, first (shorter) exam 30%, final exam 40%. There will be two non-graded exercise lists (in replacement of the non-existent exercise classes...)

Created by: Higher School of Economics

Taught by: Ekaterina Amerik, Professor
Department of Mathematics

Level	Advanced
Commitment	9 weeks of study, 4-8 hours/week
Language	English
How To Pass	Pass all graded assignments to complete the course.
User Ratings	4.3 stars Average User Rating 4.3See what learners said

Syllabus

WEEK 1

Introduction

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

1 video, 2 readings

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

Graded: Quiz 1

WEEK 2

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

Graded: QUIZ 2

WEEK 3

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 separa...

6 videos, 1 reading

Graded: QUIZ 3

WEEK 4

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

Graded: QUIZ 4

WEEK 5

Week 5

We apply the discussion from the last lecture to the case of field extensions. We show that the separable extensions remain reduced after a base change: the inseparability is responsible for eventual nilpotents. As our next subject, we introduce normal and Gal...

6 videos

Graded: Graded assignment 1

Graded: QUIZ 5

WEEK 6

Week 6

We state and prove the main theorem of these lectures: the Galois correspondence. Then we start doing examples (low degree, discriminant, finite fields, roots of unity).

6 videos

Graded: QUIZ 6

WEEK 7

Week 7

We continue to study the examples: cyclotomic extensions (roots of unity), cyclic extensions (Kummer and Artin-Schreier extensions). We introduce the notion of the composite extension and make remarks on its Galois group (when it is Galois), in the case when t...

7 videos, 1 reading

WEEK 8

Week 8

We finally arrive to the source of Galois theory, the question which motivated Galois himself: which equation are solvable by radicals and which are not? We explain Galois' result: an equation is solvable by radicals if and only if its Galois group is solvabl...

6 videos

Graded: QUIZ 8

WEEK 9

Week 9.

We build a tool for finding elements in Galois groups, learning to use the reduction modulo p. For this, we have to talk a little bit about integral ring extensions and also about norms and traces.This week, the final graded assignment is given.

6 videos

Graded: Graded assignment 2 (final exam)

Graded: QUIZ 9

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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 communications, IT, mathematics, engineering, and more. Learn more on www.hse.ru

Ratings and Reviews

Rated 4.3 out of 5 of 73 ratings

Difficult class, but well worth the effort!

The course is really well done, with good and clear videos and sets of weekly exercises. A bit concise and require a good knowledge of basic Algebra and some expansion with chapter books/notes during the lectures to fix concepts and see more examples. A set of slides could have been useful but there's a recap available from a student in the course that does a relatively good job for this. For me Week 4 and part of Week 5 (tensor products and related stuff) were really hard to follow without a previous knowledge of the matter, and I expanded it reading some suggested chapter books and internet notes (particularly useful those by Keith Conrad at U.Conn. on tensor product). I will move to five stars if some more time will be dedicated to explain tensor products and apps and make handouts.

Very interesting course where I got a lot

Work hard, learn more.

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Introduction to Galois Theory

Introduction to Galois Theory

Introduction to Galois Theory