Algebra

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. 

Information for students with disabilities and/or DSA. To ensure equal opportunities and in 
compliance with current laws, interested students may request a personal interview in order 
to plan any compensatory and/or dispensatory measures, based on their educational objectives 
and specific needs. Students with disabilities and/or DSA must contact the teacher, the CInAP 
contact person of the DMI (Prof. Daniele) and CInAP well in advance of the exam date to 
communicate that they intend to take the exam using the appropriate compensatory measures 
(which will be indicated by CInAP).

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

Highly recommended. Attending lessons and exercises allows the student to integrate the
 theory presented in the reference texts and to learn how to correctly set up the exercises
independently.

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

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

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

- then, two tests will be carried out, one in progress and one at the end of the course 
(in conjunction with the exam sessions) which, if passed, will give the student exemption 
from the written exam test.

The grade will take into account the written test (or the tests in progress) and the oral test; 
the written tests are considered passed if a grade of no less than 15/30 is obtained. 
The final grade does not consist of an average of the grades of the tests, but the oral 
determines an increase in the grade of the written test. The grade may also take into account
 any positive feedback in the tests carried out during the year. 

The learning assessment may also be carried out electronically, if conditions require it.