# Algebra

Academic Year 2023/2024 - Teacher: VINCENZO MICALE

## Expected Learning Outcomes

```The purpose of the course is to make students acquire the ability to formalize a problem and to probe the environment in which to look for any solutions. The course also aims to stimulate the ability of abstraction and provide the tools to use this abstraction to move from the particular to the general.

In particular, the course aims to make students acquire the following skills:

Knowledge and understanding: understanding statements and proofs of fundamental theorems of algebra; develop mathematical skills in reasoning, manipulation and calculation; solve mathematical problems which, although not common, are of a similar nature to others already known.

Applying knowledge and understanding: to demonstrate algebraic results not identical to those already known, but clearly related to them; build rigorous proofs; solve algebra problems that require original thinking; be able to mathematically formalize problems of moderate difficulty, formulated in natural language, and to take advantage of this formulation to clarify or solve them;
Making judgments: acquiring a conscious autonomy of judgment with reference to the evaluation and interpretation of the resolution of an algebraic problem; be able to construct and develop logical arguments with a clear identification of assumptions and conclusions; be able to recognize correct proofs, and to identify fallacious reasoning.

Communication skills: knowing how to communicate information, ideas, problems, solutions and their conclusions in a clear and unambiguous way, as well as the knowledge and rationale underlying them; know how to present scientific materials and arguments, orally or in writing, in a clear and understandable way.

Learning skills: having developed a greater degree of autonomy in the study.```

## Course Structure

```The course consists of lectures, frontal exercises (on the blackboard) and class exercises. Normally the exercises carried out by the teacher alternate with the theoretical part, even on the same day. For class exercises, the teacher proposes some exercises to the students who are invited to solve them by working in small groups; the teacher passes between the desks helping and suggesting the way to face the exercises. These exercises are essential for acquiring self-employment and group work skills.

If the teaching is given in a mixed or remote mode, the necessary changes with respect to what was previously stated may be introduced, in order to respect the program envisaged and reported in the syllabus.Information for students with disabilities and / or SLD. To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of our Department, prof. Filippo Stanco```

## Required Prerequisites

`Basic knowledge of mathematics present in all high school programs.`

## Attendance of Lessons

```The course consists of lectures, frontal exercises (on the blackboard) and class exercises. Normally the exercises carried out by the teacher alternate with the theoretical part, even on the same day. For class exercises, the teacher proposes some exercises to the students who are invited to solve them by working in small groups; the teacher passes between the desks helping and suggesting the way to face the exercises. These exercises are essential for acquiring self-employment and group work skills.

If the teaching is given in a mixed or remote mode, the necessary changes with respect to what was previously stated may be introduced, in order to respect the program envisaged and reported in the syllabus.```

## Detailed Course Content

First part (about one third of the course)

a) Elementary set theory.

Sets and operations between sets. Functions. Relations. Equivalence relations. Order relations.

b) Numbers.

Natural numbers.Induction.

Cardinality. Numeralbe sets. |A| < |P(A)|=|2A|. Not numerable sets.

Integers. Greatest common divisor and euclidean algorithm. Bézout identity. Factorization in Z and some consequences. Rational numbers.

Congruence classes. Divisibility criterions. Linear congruences. Euler function and Euler-Fermat theorem.

Real numebres as an ordered field. Complex numbers. Roots of a complex number.

Second part: algebraic structures theory.

a) Ring theory (about one third of the course)

First definitions and examples. Integral domains and fields. Subrings. Homomorphisms. Ideals. Quotients. Homomorphism theorems. Ideal generated by a subset. Prime and maximal ideals. Embedding of a domain in a field and the filed of fractions. Polynomial rings. Polynomial functions and polynomials. Ruffini theorem. Euclidean domains, PID, UFD and relations between these classes. Division between polynomials over a field. Prime and irreducible elements. Bézout identity. GCD and mcm. Gauss lemma and Gauss theorem for A[X], with A UFD. Irreducibility in A[X]. Eisenstein criterion. Irreducibility passing to quotients.

b) Groups theory (about one third of the course)

First definitions and examples. Subgroups. Cyclic groups. Permutations groups. Lagrange theorem. Normal subgroups and quotients. Homomorphisms and related theorems. Cayley's theorem. Action of a group on a set: orbits and stabilizator. Coniugacy classes. Cauchy theorem and Sylow's theorems. Direct sum of groups. Classifications theorem for finite abelian groups.

## Textbook Information

1. G. Piacentini Cattaneo - Algebra - Zanichelli.

2. A. Ragusa - Corso di Algebra (Un approccio amichevole) - Aracne Ed.

3. M. Fontana - S. Gabelli - Insiemi numeri e polinomi - CISU

## Learning Assessment

### Learning Assessment Procedures

```The final exam will consist of a written and an oral test, but it will also take into account what the student did during the year:

- during the year some exercises will be held, in which students will be offered problems to be solved individually or in small groups and during which the teacher will verify the progress of the test, suggesting ideas and correcting any errors. It will also be possible to propose tests on the theory studied.

- two tests will then take place, one in itinere and one at the end of the course (in conjunction with the exams) which, if passed, will give the student exemption from the written exam.

The grade will take into account the written test (or the ongoing tests) and the oral one; the written tests are considered passed if a grade of not less than 15/30 is obtained. The final grade does not consist of an average of the marks in the tests, but the oral determines an increase in the grade of the written test. The vote may also take into account any positive feedback in the checks carried out during the year.

Verification of learning can also be carried out electronically, should the conditions require it.```