# ALGEBRA

**Academic Year 2016/2017**- 1° Year

**Teaching Staff:**

**Marco D'ANNA**

**Credit Value:**15

**Scientific field:**MAT/02 - Algebra

**Taught classes:**84 hours

**Term / Semester:**2°

## Learning Objectives

The student of this course will be able to formalize a problem and uderstand the setting where to look for possible solutions. The student will also learn to make abstract argomentations and how to translate a particular problem in a more general setting.

## Detailed Course Content

**First part (about one third of the course)**

**a) Elementary set theory.**

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

**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.

**c) I polinomi. **

Polynomial and polynomial functions. Division algorithmo. GCD and Bézout identity. Ruffini's theorem. Unique factorization. Irreducibility in C[x], R[x], Q[x], Z[x].

**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. Euclidean domains, PID, UFD and relations between these classes. Prime and irreducible elements. Bézout identity. GCD and mcm. Gauss lemma and Gauss theorem for A[X], with A UFD. Irreducibility in A[X].

**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.