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Aprox. 45 horas para completar

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

Inglés (English)

Subtítulos: Inglés (English)

100 % en línea

Comienza de inmediato y aprende a tu propio ritmo.

Fechas límite flexibles

Restablece las fechas límite en función de tus horarios.

Nivel avanzado

Aprox. 45 horas para completar

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

Inglés (English)

Subtítulos: Inglés (English)

Programa - Qué aprenderás en este curso

Semana
1
23 minutos para completar

Introduction

This is just a two-minutes advertisement and a short reference list....
1 video (Total 3 minutos), 2 readings
2 lecturas
Introduction/Manual10m
References10m
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....
6 videos (Total 84 minutos), 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 ejercicio de práctica
Quiz 140m
Semana
2
2 horas 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....
5 videos (Total 67 minutos), 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 ejercicio de práctica
QUIZ 240m
Semana
3
4 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. ...
6 videos (Total 82 minutos), 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 lectura
Ungraded assignment 1s
1 ejercicio de práctica
QUIZ 340m
Semana
4
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")....
6 videos (Total 91 minutos), 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 ejercicio de práctica
QUIZ 440m
Semana
5
4 horas para completar

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 minutos), 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 ejercicio de práctica
QUIZ 540m
Semana
6
2 horas para completar

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 minutos), 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 ejercicio de práctica
QUIZ 640m
Semana
7
4 horas para completar

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 minutos), 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 lectura
Ungraded assignment 25m
Semana
8
2 horas para completar

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 minutos), 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 ejercicio de práctica
QUIZ 840m
Semana
9
4 horas para completar

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 minutos), 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 ejercicio de práctica
QUIZ 940m
4.3
27 revisionesChevron Right

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

Avatar

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 communicamathematics, engineering, and more. Learn more on www.hse.ru...

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