OVERVIEW

Algebra 1 presents the basic mathematical language and a first introduction to algebraic structures. This is done through the preliminary analysis of the algebraic structures of the set of integers and of the set of polynomials with coefficients in a field, of their quotients and through the study of the first properties of abstract groups.

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of this course is to provide students with basic mathematical language, to introduce them to more abstract algebraic notions by first studying properties of the integers, of univariate polynomials with coefficeints in rational, real, complex numbers or in finite fields, and of quotients of polynomial rings, to introduce them to basic aspects of group theory.

AIMS AND LEARNING OUTCOMES

Algebra 1 aims to provide the basics of:
1) mathematical language and formalization.
2) concrete algebraic structures. In particular those derived from integers and from the set of polynomials.
3) abstract algebraic structures. In particular, integer and polynomial quotients and basic notions of group theory.
The expected learning outcomes are:
1) At the end of Algebra 1 one is able to understand and write sentences using formal mathematical language.
2) At the end of Algebra 1 one is able to solve exercises related to applications between sets, equivalence relations, cardinality.
3) At the end of Algebra 1 one is able to compare and classify concrete algebraic structures arising from integers and polynomials.
4) At the end of Algebra 1 one is able to answer questions about  the structure of an abstract group and its quotients,
5) At the end of Algebra 1 one is able to reproduce, analyze and generalize the main proofs presented.

TEACHING METHODS

Standard blackboard lectures and tutorial sections.

SYLLABUS/CONTENT

- The language of mathematics. Sets and applications. Surjective, injective and bijective maps.

- Binary operations and their properties. Equivalence relations, quotient sets.

- Cardinality, countable and uncountable sets. Induction.

- Permutations, Newton's binomial and basic combinatorial notions.

- Integers: Euclidean algorithm and applications. Prime numbers and unique factorization. Fundamental theorem of arithmetic. Zero-divisor, invertible and nilpotent elements in modular algebras. Chinese remainder theorem.

- Complex numbers.

- Polynomials: polynomials in one variable with rational, real, complex and finite field coefficients. Unique factorization for polynomials. Irreducibility criteria. Quotients, zero-divisors, invertibles and nilpotents. Chinese remainder theormem in for polynomials.

- Introduction to abstract algebraic structures. Groups, period, subgroups, normal subgroups, homomorphisms and quotients. Lagrange theorem.

RECOMMENDED READING/BIBLIOGRAPHY

Luca Barbieri-Viale, "Che cosa e' un numero?", Cortina Ed. 2013.

Lindsay N. Childs, "Algebra, un'introduzione concreta", (traduzione di Carlo Traverso), ETS Editrice Pisa, 1989.

M. Artin, Algebra, Bollati Boringhieri

I. N. Herstein, Algebra, Editori Riuniti

Written notes in italian will be available via aula-web

TEACHERS AND EXAM BOARD

Exam Board

ALDO CONCA (President)

ALESSANDRO DE STEFANI

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of a written test and an oral exam.

Exam schedule

FURTHER INFORMATION

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the office Settore Servizi di supporto alla disabilità e agli studenti con DSA and the teachers  at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools. Good afternoon !