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Introduction to Galois Theory

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Course Summary

Learn Galois theory, the study of field extensions and their automorphisms, through a series of examples and exercises.

Key Learning Points

Understand the fundamental concepts of Galois theory
Apply Galois theory to solve problems in algebra and number theory
Learn through examples and exercises

Job Positions & Salaries of people who have taken this course might have

Mathematician
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- India: ₹1,233,000
- Spain: €38,000
Data Analyst
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- India: ₹550,000
- Spain: €24,000
Software Engineer
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Learning Outcomes

Understand the core concepts of Galois theory
Be able to apply Galois theory to solve problems in algebra and number theory
Gain a deeper understanding of the beauty and elegance of mathematics

Prerequisites or good to have knowledge before taking this course

A good understanding of algebra and number theory
Familiarity with mathematical proofs

Course Difficulty Level

Intermediate

Course Format

Online
Self-paced
Video lectures
Problem sets

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Description

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...) Do you have technical problems? Write to us: coursera@hse.ru

Outline

Introduction
About the University
Introduction to Galois Theory
About University
Rules on the academic integrity in the course
Introduction/Manual
References

Week 1
1.1 Field extensions: examples
1.2 Algebraic elements. Minimal polynomial.
1.3 Algebraic elements. Algebraic extensions.
1.4 Finite extensions. Algebraicity and finiteness.
1.5 Algebraicity in towers. An example.
1.6. A digression: Gauss lemma, Eisenstein criterion.
Quiz 1

Week 2
2.1 Stem field. Some irreducibility criteria.
2.2 Splitting field.
2.3 An example. Algebraic closure.
2.4 Algebraic closure (continued).
2.5 Extension of homomorphisms. Uniqueness of algebraic closure.
QUIZ 2

Week 3
3.1 An example (of extension). Finite fields.
3.2 Properties of finite fields.
3.3 Multiplicative group and automorphism group of a finite field.
3.4 Separable elements.
3.5. Separable degree, separable extensions.
3.6 Perfect fields.
Ungraded assignment 1
QUIZ 3

Week 4
4.1 Definition of tensor product
4.2 Tensor product of modules
4.3 Base change
4.4 Examples. Tensor product of algebras.
4.5 Relatively prime ideals. Chinese remainder theorem.
4.6 Structure of finite algebras over a field. Examples.
QUIZ 4

Week 5
5.1 Structure of finite K-algebras, examples (cont'd)
5.2 Separability and base change
5.3 Separability and base change (cont'd). Primitive element theorem.
5.4 Examples. Normal extensions.
5.5 Galois extensions.
5.6 Artin's theorem.
QUIZ 5

Week 6
6.1 Some further remarks on normal extension. Fixed field.
6.2 The Galois correspondence
6.3 Galois correspondence (cont'd). First examples (polynomials of degree 2 and 3.
6.4 Discriminant. Degree 3 (cont'd). Finite fields.
6.5 An infinite degree example. Roots of unity: cyclotomic polynomials
6.6 Irreducibility of cyclotomic polynomial.The Galois group.
QUIZ 6

Week 7
7.1 Cyclotomic extensions (cont'd). Examples over Q.
7.2. Kummer extensions.
7.3. Artin-Schreier extensions.
7.4. Composite extensions. Properties.
7.5. Linearly disjoint extensions. Examples.
7.6. Linearly disjoint extensions in the Galois case.
7.7 On the Galois group of the composite.
Ungraded assignment 2

Week 8
8.1. Extensions solvable by radicals. Solvable groups. Example.
8.2. Properties of solvable groups. Symmetric group.
8.3.Galois theorem on solvability by radicals.
8.4.Examples of equations not solvable by radicals."General equation".
8.5. Galois action as a representation. Normal base theorem.
8.6. Normal base theorem (cont'd). Relation with coverings.
QUIZ 8

Week 9.
9.1 Integral elements over a ring.
9.2. Integral extensions, integral closure, ring of integers of a number field.
9.3. Norm and trace.
9.4. Norm and trace (cont'd). Ring of integers is a free module.
9.5. Reduction modulo a prime.
9.6. Reduction modulo a prime and finding elements in Galois groups.
QUIZ 9

Summary of User Reviews

Discover the beauty of abstract algebra with Galois theory! This course has received numerous positive reviews from learners who found it to be an excellent introduction to the subject. Many users appreciated the clear explanations and insightful examples provided by the instructor.

Key Aspect Users Liked About This Course

Clear explanations and insightful examples provided by the instructor.

Pros from User Reviews

Excellent introduction to abstract algebra
Engaging and well-structured course material
Challenging exercises to reinforce learning
Great support from the instructor and community
Opportunity to apply concepts to real-world problems

Cons from User Reviews

Some learners found the pace too fast
Not enough emphasis on practical applications
Requires a strong mathematical background
Lack of visual aids or interactive elements
Limited opportunities for interaction with other learners

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