Introduction to Galois Theory
National Research University Higher School of Economics
课程信息
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...)
Syllabus - What you will learn from this course
1
Section
Introduction
This is just a two-minutes advertisement and a short reference list.
1 video (Total 3 min), 2 readings
1 videos
2 readings
Introduction/Manual10m
References10m
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 min), 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 practice exercises
Quiz 112m
2
Section
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 min), 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 practice exercises
QUIZ 212m
3
Section
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 min), 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 readings
Ungraded assignment 110m
1 practice exercises
QUIZ 38m
4
Section
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 min), 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 practice exercises
QUIZ 410m
5
Section
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 min), 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 practice exercises
QUIZ 510m
6
Section
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 min), 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 practice exercises
QUIZ 68m
7
Section
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 min), 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 readings
Ungraded assignment 210m
8
Section
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 min), 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 practice exercises
QUIZ 88m
9
Section
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 min), 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 practice exercises
QUIZ 98m
4.3
Top Reviews
By 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.
By HC•Feb 27th 2017
An intriguing course.\n\nUncommonly advanced material.\n\nRare content on Coursera.
Instructor
Ekaterina Amerik
ProfessorAbout 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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