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.

In the course the will be lectures and exercises, given at the blackboard by the teacher, and class exercises. Ussually the lecturer alternates exercises and theoretical parts in the same day. As for class exercises, the lecturer gives some exercises to the students, that have to try to solve them working in small groups; the lecturer helps the students to find the proper way to appoach the exercises. Together with the lecturers of Analysis and Geometry some joint lectures will take place, in order to let the students better understand the connetions between these subjects and the unity of mathematical knowledge.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

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.

​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