Università di Pavia - Offerta formativa

ALGEBRA 2

Enrollment year

2019/2020

Academic year

2020/2021

Regulations

DM270

Academic discipline

MAT/02 (ALGEBRA)

Department

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

Course

MATHEMATICS

Curriculum

PERCORSO COMUNE

Year of study

2°

Period

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

ECTS

6

Lesson hours

56 lesson hours

Language

Italian

Activity type

WRITTEN AND ORAL TEST

Teacher

CANONACO ALBERTO (titolare) - 6 ECTS

Prerequisites

The courses of Linear algebra and Algebra 1.

Learning outcomes

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

Course contents

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.

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.

Teaching methods

Lectures and exercise sessions

Reccomended or required readings

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.

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.

Assessment methods

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.

Further information

Sustainable development goals - Agenda 2030