
Course unit
GALOIS THEORY
SCM0014410, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2017/18
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/02 
Algebra 
7.0 
Course unit organization
Period 
First semester 
Year 
3rd Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
3.0 
24 
51.0 
No turn 
Lecture 
4.0 
32 
68.0 
No turn 
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 (and their proofs), encountered in the course. 
Assessment criteria:

The knowledge and the ability to apply the notions and results seen during the course will be evaluated. 
Course unit contents:

Polynomials and their roots. Artin theorem on simple extensions. Separable and purely inseparable extensions of fields. Splitting fields. Algebraic closure of a field. Galois extensions. Cyclotomic Extensions. Jordan Holder Theorem. Soluble groups. Fundamental theorem of algebra. Resolubility for radicals. Galois Theorem. Berlekamp algorithm. Cyclic extensions. 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) 

D.J.H. Garling, A course in Galois Theory. : Cambridge University Press 1986, .

J.S. Milne, Fields and Galois Theory. : (note disponibili in rete), .

I. Martin Isaacs, Algebra, a graduate course. : AMS, .


