ALGEBRA

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.

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.

1. G. Piacentini Cattaneo - Algebra - Zanichelli.

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