First Level Degree in Mathematics
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GALOIS THEORY - 7 CFU
Teacher
Riccardo Colpi
Scheduled Period
III Year - 1 Semester | 04/10/2021 - 15/01/2022
Hours: 56 (24 exercises, 32 lesson)
Prerequisites
Algebra and Geometry courses of the first and second years: in particular groups, rings fields and linear algebra.
Target skills and knowledge
The classical theory of the fields and the theory of Galois will be presented. In particular: ruler and compass constructions, solubility for radicals of algebraic equations, field extensions, normality, separability.
Examination methods
Written and oral exams. In the written exam, the student must demonstrate to be able to solve exercises of the Galois theory. The oral exam, in which the vote is decided, is dedicated to verify the knowledge of the definitions and the results encountered in the course.
Assessment criteria
The knowledge and the ability to apply the notions and results contained in the course will be evaluated.
Course contents
Polynomials and their roots. Artin theorem on simple extensions. Irreducibility criteria. Separable and inseparable extensions. Splitting fields. Algebraic closure of a field. Galois extensions and fundamental theorem. Cyclotomic extensions and radical extensions. Jordan Holder Theorem. Soluble groups. Resolubility by radicals. Galois Theorem. Primitive elements. Dedekind's theorem. Ruler and compass constructions. Galois groups of polynomials up to the fourth degree.
Planned learning activities and teaching methods
Frontal lessons, using a tablet.
Additional notes about suggested reading
The study material is made up of suggested text books, lesson notes, and any other notes that will be made available on the course website.
Textbooks (and optional supplementary readings)
- J.S. Milne, Fields and Galois Theory, (note disponibili in rete)
- D.J.H. Garling, A course in Galois Theory, Cambridge University Press 1986.
- I. Martin Isaacs, Algebra, a graduate course, AMS.