#
Algebra 1

A.Y. 2021/2022

Learning objectives

The aim of the course is to provide an Introduction to Mathematics and its fundamental constructions: numbers and algebraic structures. The construction of the set of natural numbers is made via a set-theoretical approach. The concept of algebraic structure is also central: through it, several constructions of sets of numbers as, for instance, integers and rationals, are characterized.

Expected learning outcomes

Knowledge of the basics of Algebra, of the main abstract algebraic structures through the study of integers, polynomials in one variable and their factor rings. Finally, the basic properties of modules are treated.

**Lesson period:**
First semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Algebra 1 (ediz.)

Responsible

Lesson period

First semester

More specific information on the delivery modes of training activities for academic year 2021/22 will be provided over the coming months, based on the evolution of the public health situation.

**Course syllabus**

Basics on sets, binary relations and functions. Cantor's Theorem. Natural numbers, Peano Axioms and induction. Recursivity. Finite and infinite sets. Axiom of Choice. Equivalence relations, partitions and quotient sets. Countable sets and cardinality. Construction of integer and rational numbers. Order relations. Lattices. Prime numbers. Greatest common divisor and least common multiple. Euclidean algorithm and its applications. Unique factorization in product of primes. Congruences. Modular algebra and diophantine equations. Chinese Remainder Theorem. Little Fermat's Theorem and Euler function. Binary operations and their properties: fields, rings, groups, semigrous and monoids. Rings, subrings and ideals. Homomorphisms, factor rings and kernels. Characteristic of a ring. Polynomial rings. Irreducibility and unique factorization. Unique factorization domains (UFD's) and principal ideals domains (PID's). Euclidean domains. Subrings of the complex field and Gauss integers. Polynomials with integral coefficients and Gauss Lemma. Modules and algebras: finitely generated modules and free modules. Bases and rank of a module.

**Prerequisites for admission**

No specific mathematical background is required.

**Teaching methods**

Traditional lectures, exercise classes.

**Teaching Resources**

L. Barbieri Viale: Che cos'è un numero? Un'introduzione all'Algebra. Raffaello Cortina, 2013

**Assessment methods and Criteria**

The final examination consists of two parts: a written exam and an oral exam.

- During the written exam, the student must solve some exercises in the format of open-ended questions, with the aim of assessing the student's ability to solve problems using the techniques described in the course. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first or the second examination session, the student may choose instead to take two midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.

- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if all two parts (written, oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.

- During the written exam, the student must solve some exercises in the format of open-ended questions, with the aim of assessing the student's ability to solve problems using the techniques described in the course. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first or the second examination session, the student may choose instead to take two midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.

- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if all two parts (written, oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.

MAT/02 - ALGEBRA - University credits: 9

Practicals: 48 hours

Lessons: 45 hours

Lessons: 45 hours

Professors:
Barbieri Viale Luca, Vezzani Alberto

### Algebra 1 (ediz.2)

Responsible

Lesson period

First semester

More specific information on the delivery modes of traininig activities for the academic year 20021/2022 will be provided in the next few months, based on the evolution of the public health situation.

**Course syllabus**

Basics on sets, binary relations and functions. Cantor's Theorem. Natural numbers, Peano Axioms and induction. Recursivity. Finite and infinite sets. Axiom of Choice. Equivalence relations, partitions and quotient sets. Countable sets and cardinality. Construction of integer and rational numbers. Order relations. Lattices. Prime numbers. Greatest common divisor and least common multiple. Euclidean algorithm and its applications. Unique factorization in product of primes. Congruences. Modular algebra and diophantine equations. Chinese Remainder Theorem. Little Fermat's Theorem and Euler function. Binary operations and their properties: fields, rings, groups, semigrous and monoids. Rings, subrings and ideals. Homomorphisms, factor rings and kernels. Characteristic of a ring. Polynomial rings. Irreducibility and unique factorization. Unique factorization domains (UFD's) and principal ideals domains (PID's). Euclidean domains. Subrings of the complex field and Gauss integers. Polynomials with integral coefficients and Gauss Lemma. Modules and algebras: finitely generated modules and free modules. Bases and rank of a module.

**Prerequisites for admission**

No specific mathematical background is required.

**Teaching methods**

Traditional blackboard lectures, exercise classes.

**Teaching Resources**

L. Barbieri Viale: Che cos'è un numero? Un'introduzione all'Algebra. Raffaello Cortina, 2013

D.Dikranjan, M.S.Lucido: Aritmetica e Algebra. Liguori, 2007

M. Curzio, P. Longobardi, M. Maj: Lezioni di Algebra. Liguori, 1994

D.Dikranjan, M.S.Lucido: Aritmetica e Algebra. Liguori, 2007

M. Curzio, P. Longobardi, M. Maj: Lezioni di Algebra. Liguori, 1994

**Assessment methods and Criteria**

The final examination consists of two parts: a written exam and an oral exam.

- During the written exam, the student must solve some exercises in the format of open-ended questions, with the aim of assessing the student's ability to solve problems using the techniques described in the course. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first or the second examination session, the student may choose instead to take two midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.

- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if all two parts (written, oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.

- During the written exam, the student must solve some exercises in the format of open-ended questions, with the aim of assessing the student's ability to solve problems using the techniques described in the course. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first or the second examination session, the student may choose instead to take two midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.

- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if all two parts (written, oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.

MAT/02 - ALGEBRA - University credits: 9

Practicals: 48 hours

Lessons: 45 hours

Lessons: 45 hours

Professors:
Andreatta Fabrizio, Mazza Carlo

Professor(s)