CODE 25897 2024/2025 9 cfu anno 1 MATEMATICA 8760 (L-35) - GENOVA 9 cfu anno 1 STATISTICA MATEM. E TRATTAM. INFORMATICO DEI DATI 8766 (L-35) - GENOVA MAT/02 Italian GENOVA 1° Semester AULAWEB

## 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 at providing the basics of:
1) Mathematical language and formalization.
2) Concrete algebraic structures. In particular those derived from the set of integers and 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 in class.

### TEACHING METHODS

Standard blackboard lectures, exercises 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, subgroups, cyclic groups and order of an element. Normal subgroups, homomorphisms and quotients. Lagrange theorem.

Notes in italian, made available via Aulaweb

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

## LESSONS

### LESSONS START

The class will start accordingly with 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. The written test can also be passed by taking two midterm tests (called "compitini"), one in the middle and one at the end of the semester.

### ASSESSMENT METHODS

In the written test students face problems at different levels of difficulty. Some of them are reproductions of questions seen in class while others need an individual elaboration process, starting from concepts tackled in class.

In the oral test students discuss exercises and reproduce the main steps of the theoretical notions seen in class, including presentation, analysis and generalization of the main proofs.

The evaluation will be based both on the level of knowledge of the topics, and on the ability of analyze and formalize concepts in a correct mathematical language.

The final grade is based on the grade obtained on the written exam, and can change depending how the student performs at the oral exam.

### FURTHER INFORMATION

Attendance is not mandatory, but strongly recommended.

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs must follow the instructions available on Aulaweb at https://2023.aulaweb.unige.it/course/view.php?id=12490#section-3 in order to agree on special arrangements.

Requests should be sent well in advance (at least 10 days) before the date of the exam by sending an email to the professor with the "Referente di Scuola" and the DSA office in Cc (please see instructions).