SEMANA 1

Introduction

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

1 vídeo, 2 leituras

Reading: Introduction/Manual
Reading: References

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 vídeos

Vídeo: 1.2 Algebraic elements. Minimal polynomial.
Vídeo: 1.3 Algebraic elements. Algebraic extensions.
Vídeo: 1.4 Finite extensions. Algebraicity and finiteness.
Vídeo: 1.5 Algebraicity in towers. An example.
Vídeo: 1.6. A digression: Gauss lemma, Eisenstein criterion.

Nota atribuída: Quiz 1

SEMANA 2

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 vídeos

Vídeo: 2.2 Splitting field.
Vídeo: 2.3 An example. Algebraic closure.
Vídeo: 2.4 Algebraic closure (continued).
Vídeo: 2.5 Extension of homomorphisms. Uniqueness of algebraic closure.

Nota atribuída: QUIZ 2

SEMANA 3

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 vídeos, 1 leitura

Vídeo: 3.2 Properties of finite fields.
Vídeo: 3.3 Multiplicative group and automorphism group of a finite field.
Vídeo: 3.4 Separable elements.
Vídeo: 3.5. Separable degree, separable extensions.
Vídeo: 3.6 Perfect fields.
Reading: Ungraded assignment 1

Nota atribuída: QUIZ 3

SEMANA 4

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 vídeos

Vídeo: 4.2 Tensor product of modules
Vídeo: 4.3 Base change
Vídeo: 4.4 Examples. Tensor product of algebras.
Vídeo: 4.5 Relatively prime ideals. Chinese remainder theorem.
Vídeo: 4.6 Structure of finite algebras over a field. Examples.

Nota atribuída: QUIZ 4

SEMANA 5

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 vídeos

Vídeo: 5.2 Separability and base change
Vídeo: 5.3 Separability and base change (cont'd). Primitive element theorem.
Vídeo: 5.4 Examples. Normal extensions.
Vídeo: 5.5 Galois extensions.
Vídeo: 5.6 Artin's theorem.

Nota atribuída: Graded assignment 1

Nota atribuída: QUIZ 5

SEMANA 6

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 vídeos

Vídeo: 6.2 The Galois correspondence
Vídeo: 6.3 Galois correspondence (cont'd). First examples (polynomials of degree 2 and 3.
Vídeo: 6.4 Discriminant. Degree 3 (cont'd). Finite fields.
Vídeo: 6.5 An infinite degree example. Roots of unity: cyclotomic polynomials
Vídeo: 6.6 Irreducibility of cyclotomic polynomial.The Galois group.

Nota atribuída: QUIZ 6

SEMANA 7

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 vídeos, 1 leitura

Vídeo: 7.2. Kummer extensions.
Vídeo: 7.3. Artin-Schreier extensions.
Vídeo: 7.4. Composite extensions. Properties.
Vídeo: 7.5. Linearly disjoint extensions. Examples.
Vídeo: 7.6. Linearly disjoint extensions in the Galois case.
Vídeo: 7.7 On the Galois group of the composite.
Reading: Ungraded assignment 2

SEMANA 8

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 vídeos

Vídeo: 8.2. Properties of solvable groups. Symmetric group.
Vídeo: 8.3.Galois theorem on solvability by radicals.
Vídeo: 8.4.Examples of equations not solvable by radicals."General equation".
Vídeo: 8.5. Galois action as a representation. Normal base theorem.
Vídeo: 8.6. Normal base theorem (cont'd). Relation with coverings.

Nota atribuída: QUIZ 8

SEMANA 9

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 vídeos

Vídeo: 9.2. Integral extensions, integral closure, ring of integers of a number field.
Vídeo: 9.3. Norm and trace.
Vídeo: 9.4. Norm and trace (cont'd). Ring of integers is a free module.
Vídeo: 9.5. Reduction modulo a prime.
Vídeo: 9.6. Reduction modulo a prime and finding elements in Galois groups.

Nota atribuída: Graded assignment 2 (final exam)

Nota atribuída: QUIZ 9

Nível	Advanced
Compromisso	9 weeks of study, 4-8 hours/week
Idioma	English
Como ser aprovado	Seja aprovado em todas as tarefas para concluir o curso.
Classificação do usuário	4.3 estrelas Classificação média do usuário 4.3Veja o que os aprendizes disseram

Introduction to Galois Theory

Introduction to Galois Theory