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## BASICS OF HIGHER ALGEBRA

CODE 90694 2020/2021 7 credits during the 1st year of 9011 Mathematics (LM-40) GENOVA 7 credits during the 3nd year of 8760 Mathematics (L-35) GENOVA MAT/02 Italian (English on demand) GENOVA (Mathematics) 2° Semester AULAWEB

## 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.

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

## TEACHERS AND EXAM BOARD

### Exam Board

ALDO CONCA (President)

FRANCESCO VENEZIANO

ANNA MARIA BIGATTI (President Substitute)

EMANUELA DE NEGRI (President Substitute)

ALESSANDRO DE STEFANI (President Substitute)

MARIA EVELINA ROSSI (President Substitute)

MATTEO VARBARO (President Substitute)

## 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.

### Class schedule

BASICS OF HIGHER ALGEBRA

## EXAMS

### EXAM DESCRIPTION

Oral and computer algebra  project