Acerca de este Curso
4.3
90 ratings
26 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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Advanced Level

Nivel avanzado

Clock

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

Aprox. 43 horas para completar
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English

Subtítulos: English
Globe

Cursos 100 % en línea

Comienza de inmediato y aprende a tu propio ritmo.
Calendar

Fechas límite flexibles

Restablece las fechas límite en función de tus horarios.
Advanced Level

Nivel avanzado

Clock

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

Aprox. 43 horas para completar
Comment Dots

English

Subtítulos: English

Programa - Qué aprenderás en este curso

1

Sección
Clock
23 minutos para completar

Introduction

This is just a two-minutes advertisement and a short reference list....
Reading
1 video (Total: 3 min), 2 readings
Reading2 lecturas
Introduction/Manual10m
References10m
Clock
2 horas para completar

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....
Reading
6 videos (Total: 84 min), 1 quiz
Video6 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
Quiz1 ejercicio de práctica
Quiz 112m

2

Sección
Clock
1 hora para completar

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....
Reading
5 videos (Total: 67 min), 1 quiz
Video5 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
Quiz1 ejercicio de práctica
QUIZ 212m

3

Sección
Clock
2 horas para completar

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. ...
Reading
6 videos (Total: 82 min), 1 reading, 1 quiz
Video6 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
Reading1 lectura
Ungraded assignment 110m
Quiz1 ejercicio de práctica
QUIZ 38m

4

Sección
Clock
2 horas para completar

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")....
Reading
6 videos (Total: 91 min), 1 quiz
Video6 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
Quiz1 ejercicio de práctica
QUIZ 410m
4.3

Principales revisiones

por 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

Ekaterina Amerik

Professor
Department of Mathematics

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

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