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.
oferecido por
Introduction to Galois Theory
Universidade HSEInformações sobre o curso
7.970 visualizações recentes
Prazos flexíveis
Redefinir os prazos de acordo com sua programação.
Certificados compartilháveis
Tenha o certificado após a conclusão
100% on-line
Comece imediatamente e aprenda em seu próprio cronograma.
Nível avançado
Aprox. 27 horas para completar
Inglês
Legendas: Francês, Portuguese (European), Russo, Inglês, Espanhol
Prazos flexíveis
Redefinir os prazos de acordo com sua programação.
Certificados compartilháveis
Tenha o certificado após a conclusão
100% on-line
Comece imediatamente e aprenda em seu próprio cronograma.
Nível avançado
Aprox. 27 horas para completar
Inglês
Legendas: Francês, Portuguese (European), Russo, Inglês, Espanhol
oferecido por
Universidade HSE
HSE University 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.
Programa - O que você aprenderá com este curso
1 hora para concluir
Introduction
This is just a two-minutes advertisement and a short reference list.
1 hora para concluir
2 vídeos (Total 4 mín.), 4 leituras
2 horas para concluir
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.
2 horas para concluir
6 vídeos (Total 84 mín.)
2 horas para concluir
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.
2 horas para concluir
5 vídeos (Total 67 mín.)
4 horas para concluir
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.
4 horas para concluir
6 vídeos (Total 82 mín.), 1 leitura, 1 teste
2 horas para concluir
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").
2 horas para concluir
6 vídeos (Total 91 mín.)
Avaliações
Principais avaliações do INTRODUCTION TO GALOIS THEORY
por MP30 de Jul de 2020
A difficult course for me, personally, but that makes it all the more worth it! Taking this course has helped me learn more I thought I would. Definitely recommended.
por F11 de Set de 2021
Thank You Prof Ekaterina Amerik for this course. I understand with your explaination. Maybe i must study again and again for understand this material.
por CL15 de Jun de 2016
Outstanding course so far - a great refresher for me on Galois theory. It's nice to see more advanced mathematics classes on Coursera.
por TW11 de Mar de 2018
The teacher is good at explaining things. It is best you take an algebra course for prerequisite.
Perguntas Frequentes – FAQ
Quando terei acesso às palestras e às tarefas?
O que recebo ao adquirir o Certificado?
Existe algum auxílio financeiro disponível?
Mais dúvidas? Visite o Central de Ajuda ao estudante.
