ofrecido por

National Research University Higher School of Economics

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...)...

Comienza de inmediato y aprende a tu propio ritmo.

Restablece las fechas límite en función de tus horarios.

Aprox. 43 horas para completar

Subtítulos: English

Comienza de inmediato y aprende a tu propio ritmo.

Restablece las fechas límite en función de tus horarios.

Aprox. 43 horas para completar

Subtítulos: English

Sección

This 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 112m

Sección

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

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 212m

Sección

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

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 110m

QUIZ 38m

Sección

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

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 410m

4.3

por 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.

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...

When will I have access to the lectures and assignments?

Once you enroll for a Certificate, you’ll have access to all videos, quizzes, and programming assignments (if applicable). Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments.

What will I get if I purchase the Certificate?

When you purchase a Certificate you get access to all course materials, including graded assignments. Upon completing the course, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.

What is the refund policy?

Is financial aid available?

¿Tienes más preguntas? Visita el Centro de Ayuda al Alumno.

Coursera brinda acceso universal a la mejor educación del mundo, al asociarse con las mejores universidades y organizaciones, para ofrecer cursos en línea.