2019/2020

2020/2021

DM270

MAT/02 (ALGEBRA)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

2°

2nd semester (01/03/2021 - 11/06/2021)

6

56 lesson hours

Italian

WRITTEN AND ORAL TEST

CANONACO ALBERTO (titolare) - 6 ECTS

The courses of Linear algebra and Algebra 1.

The course is an introduction to Galois theory, with some complements of group theory and of the theory of modules over a ring.

Modules over a ring; submodules, module homomorphisms and quotient modules. Products and direct sums of modules; free modules. Noetherian modules; decomposability of modules; simple modules and semisimple modules. The structure theorem for finitely generated modules over a principal ideal domain.
Finitely generated abelian groups. Group actions on sets; group representations. The class equation. Cauchy theorem and Sylow theorem. Simple and soluble groups. Semidirect products of groups.
Field extensions; algebraic and transcendental elements. Ruler and compass constructions. Splitting fields of polynomials. Algebraic closure of a field. Normal, separable and Galois extensions. Fixed fields and Galois groups; the fundamental theorem of Galois theory. Galois theory for finite fields. Polynomials solvable by radicals.

Lectures and exercise sessions

Notes provided by the teacher.
I.N. Herstein, "Algebra", Editori Riuniti.
M. Artin, "Algebra", Bollati Boringhieri.
P. Aluffi, "Algebra: chapter 0", American Mathematical Society.
J.S. Milne, "Group Theory", http://www.jmilne.org/math/CourseNotes/gt.html.
D.J.H. Garling, "A Course in Galois Theory", Cambridge University Press.
I.N. Stewart, "Galois Theory", CRC Press.
J.S. Milne, "Fields and Galois Theory", http://www.jmilne.org/math/CourseNotes/ft.html.

The exam consists of a written test, during which the student must solve some exercises, and of an oral examination, during which the student must answer some questions, mainly of a theoretical nature.