Offered By

National Research University Higher School of Economics

About this Course

4.3

96 ratings

•

27 reviews

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...)

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

Subtitles: English

Start instantly and learn at your own schedule.

Reset deadlines in accordance to your schedule.

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

Subtitles: English

Week

1This is just a two-minutes advertisement and a short reference list....

1 video (Total 3 min), 2 readings

Introduction/Manual10m

References10m

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

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

Quiz 140m

Week

2We 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

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

QUIZ 240m

Week

3We 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

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

Ungraded assignment 1s

QUIZ 340m

Week

4This 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

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

QUIZ 440m

4.3

27 ReviewsBy CL•Jun 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.

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