# BASICS OF HIGHER ALGEBRA

iten
Code
90694
ACADEMIC YEAR
2017/2018
CREDITS
7 credits during the 1st year of 9011 Mathematics (LM-40) GENOVA

7 credits during the 3nd year of 8760 Mathematics (L-35) GENOVA

SCIENTIFIC DISCIPLINARY SECTOR
MAT/02
LANGUAGE
Italian (English on demand)
TEACHING LOCATION
GENOVA (Mathematics)
semester
2° Semester
Teaching materials

## OVERVIEW

Language: English

## AIMS AND CONTENT

LEARNING OUTCOMES

The goal of the course is the study of system of polynomial equations via Galois theory and using  Groebner bases.

TEACHING METHODS

Teaching style: In presence+computational sections with the use of symbolic computation packages.

SYLLABUS/CONTENT

I - Rings, ideals and modules. Noetherian rings and the Hilbert basis theorem. Polynomials: The ring K [x_1, ..., x_n] of polynomials with coefficientsin a field. Grobner Bases  and the Buchberger algorithm. Systems of polynomial equations and elimination theory.

II - Review of field extensions. Splitting fields of polynomials with coefficients in a field of characteristic 0, normal extensions and their basic properties. Fundamental Theorem of Galois theory. The Galois group of a polynomial. Applications: cyclotomic fields, solvability by radicals of algebraic equations.

RECOMMENDED READING/BIBLIOGRAPHY

Computational Commutative Algebra, Kreuzer, Robbiano, Springer, 2004.

## TEACHERS AND EXAM BOARD

Office hours: To be decided later on when the general timetable will be fixed.&nbsp;

Exam Board

ANNA MARIA BIGATTI (President)

ALDO CONCA (President)

STEFANO VIGNI (President)

## LESSONS

TEACHING METHODS

Teaching style: In presence+computational sections with the use of symbolic computation packages.

LESSONS START

The class will start according to the academic calendar.

## EXAMS

EXAM DESCRIPTION

Oral and computer algebra  project

Exam schedule

Date Time Location Type Notes
22/01/2018 09:00 GENOVA Orale
01/02/2018 09:00 GENOVA Orale
04/06/2018 09:00 GENOVA Orale
02/07/2018 09:00 GENOVA Orale
03/09/2018 09:00 GENOVA Orale