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
23 minutes 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
1.2 Algebraic elements. Minimal polynomial.12m
1.3 Algebraic elements. Algebraic extensions.14m
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
2.2 Splitting field.11m
2.3 An example. Algebraic closure.14m
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
3.2 Properties of finite fields.14m
3.3 Multiplicative group and automorphism group of a finite field.15m
3.4 Separable elements.15m
3.5. Separable degree, separable extensions.15m
3.6 Perfect fields.9m
1 reading
Ungraded assignment 1s
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
4.2 Tensor product of modules14m
4.3 Base change14m
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
Week
5
4 hours to complete

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 Galois extensions and prove Artin's theorem on invariants. This week, the first graded assignment is given....
6 videos (Total 81 min), 2 quizzes
6 videos
5.2 Separability and base change14m
5.3 Separability and base change (cont'd). Primitive element theorem.14m
5.4 Examples. Normal extensions.13m
5.5 Galois extensions.11m
5.6 Artin's theorem.13m
1 practice exercise
QUIZ 540m
Week
6
2 hours to complete

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 (Total 86 min), 1 quiz
6 videos
6.2 The Galois correspondence14m
6.3 Galois correspondence (cont'd). First examples (polynomials of degree 2 and 3.14m
6.4 Discriminant. Degree 3 (cont'd). Finite fields.15m
6.5 An infinite degree example. Roots of unity: cyclotomic polynomials14m
6.6 Irreducibility of cyclotomic polynomial.The Galois group.14m
1 practice exercise
QUIZ 640m
Week
7
4 hours to complete

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 the composed extensions are in some sense independent and one or both of them is Galois. The notion of independence is also given a precise sense ("linearly disjoint extensions"). This week, an ungraded assignment is given....
7 videos (Total 87 min), 1 reading
7 videos
7.2. Kummer extensions.14m
7.3. Artin-Schreier extensions.11m
7.4. Composite extensions. Properties.13m
7.5. Linearly disjoint extensions. Examples.15m
7.6. Linearly disjoint extensions in the Galois case.12m
7.7 On the Galois group of the composite.7m
1 reading
Ungraded assignment 25m
Week
8
2 hours to complete

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 solvable in the sense of group theory. In particular we see that the "general" equation of degree at least 5 is not solvable by radicals. We briefly discuss the relations to representation theory and to topological coverings....
6 videos (Total 81 min), 1 quiz
6 videos
8.2. Properties of solvable groups. Symmetric group.13m
8.3.Galois theorem on solvability by radicals.11m
8.4.Examples of equations not solvable by radicals."General equation".13m
8.5. Galois action as a representation. Normal base theorem.14m
8.6. Normal base theorem (cont'd). Relation with coverings.12m
1 practice exercise
QUIZ 840m
Week
9
4 hours to complete

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 (Total 84 min), 2 quizzes
6 videos
9.2. Integral extensions, integral closure, ring of integers of a number field.15m
9.3. Norm and trace.14m
9.4. Norm and trace (cont'd). Ring of integers is a free module.13m
9.5. Reduction modulo a prime.13m
9.6. Reduction modulo a prime and finding elements in Galois groups.14m
1 practice exercise
QUIZ 940m
4.3
27 ReviewsChevron Right

Top Reviews

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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