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 formalism.
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. By the end of Algebra 1, students will be able to understand and write sentences using formal mathematical language.
2. By the end of Algebra 1, students will be able to solve exercises related to applications between sets, equivalence relations, cardinality.
3. By the end of Algebra 1, students will be able to compare and classify concrete algebraic structures arising from integers and polynomials.
4. By the end of Algebra 1, students will be able to answer questions about  the structure of an abstract group and its quotients.
5. By the end of Algebra 1, students will be able to reproduce, analyze and generalize the main proofs presented in class.

PREREQUISITES

There are no specific requirements.

TEACHING METHODS

There will be blackboard lectures covering the theoretic and practical (exercises) aspects of the course, as well as tutoring sessions. Attendance is not mandatory, but strongly recommended.

SYLLABUS/CONTENT

The main topics are:

1. The language of mathematics. Sets and applications. Surjective, injective and bijective maps.
2. Binary operations and their properties. Equivalence relations, quotient sets.
3. Cardinality, countable and uncountable sets. Induction.
4. Permutations, Newton's binomial and basic combinatorial notions.
5. 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.
6. 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 theorem for polynomials.
7. Introduction to abstract algebraic structures. Groups, subgroups, cyclic groups and order of an element. Normal subgroups, homomorphisms and quotients. Lagrange theorem.

RECOMMENDED READING/BIBLIOGRAPHY

• Notes in italian, Authors: A. De Stefani and M.E. Rossi (available via AulaWeb).
• Algebra, Author: M. Artin, Bollati Boringhieri
• Algebra, Author: I. N. Herstein, Editori Riuniti

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

The dates of the academic calendar are available on the webpages of the Corso di Studi. Further information regarding the schedule is available here.

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

Compensatory and dispensatory measures Disability/Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to facilitate the examination.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the teacher in agreement with the Referee.

To take advantage of the adaptations during the examination, fill in the Adaptation request form; the request will be automatically sent by the system to the teacher in charge of the teaching, to the Contact Person of your School/Area/Department and in copy to the Sector; you will also receive a copy of the request sent by e-mail.

The adjustments available to students are as follows:

• Additional time (+30% DSA)
• Additional time (+50% disability/invalidity)
• Additional time during oral exams to organise the answer
• Calculator (programmable and graphing calculators are not allowed)
• Conceptual Maps
• Tables and/or Forms
• Take the exam in written form
• Take the exam in oral form
• Tutor reader (for written tests only)
• Tutor-writer (for written tests only)

Your request for adaptations must be submitted at least 7 working days before the scheduled exam date.

All information for students with disabilities and DSA is available on the webpage: Services for students with disabilities or DSA | UniGe | University of Genoa

Reference for inclusion: Sergio Di Domizio - sergio.didomizio@unige.it