Learning objectives
The students will learn the basic definitions, problems, and techniques in Galois theory.
Prerequisites
Algebra (groups, rings, fields) and Linear Algebra.
Course unit content
Building on the first-year studies in Algebra and Linear Algebra, this course aims to provide a thorough understanding of the fundamentals of algebraic field extensions and Galois theory, as well as an introduction to the theory of algebraic spaces.
Full programme
Review of group theory, rings, and fields. Classification of finitely generated abelian groups. Structural properties of solvable groups. Finite and algebraic field extensions. Existence theorem of the algebraic closure. Splitting fields, normal extensions. Finite fields. Inseparable extensions. Galois extensions. Cyclic, solvable, and radical extensions. Kummer theory. Galois correspondence theorem. Applications of Galois theory: Galois's proof of the Abel-Ruffini theorem and the constructibility criterion of a complex number. Introduction to algebraic spaces: Nullstellensatz; algebraic spaces and varieties. Spectrum of a ring.
Bibliography
S. Lang, Algebra, Graduate Text in Mathematics, Springer.
Teaching methods
The topics of the course will be discussed during the lectures, together with examples, applications, and numerous exercises.
Assessment methods and criteria
At the end of the course there will be an exam in two parts. The first one amounts to turning in the solutions of a number of exercises. The second one will be at the board, where the student will be asked to discuss, explain, and prove the main results of the course.
Other information
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2030 agenda goals for sustainable development
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