Mathematics

Third Part. Elements of ring theory Rings, ideals, quotients, homomorphisms, theorems of homomorphism and isomorphism, field of fractions of a domain. - Euclidean domains, principal ideal domains, unique factorization domains, Gauss integers, whole sum of two squares, prime ideals and maximal ideals; divisibility in domains; prime and irreducible elements. Rings of polynomials: universal property, polynomials with coefficients in a domain, the Euclidean property of monic polynomials, quotient rings of polynomial rings, Gaussian lemma, Eisenstein criterion and other irreducibility criteria; the irreducible elements of Z[x], unique factorization in Z[x]. Fourth part. Elements of field theory Extensions of fields, algebraic and transcendental elements, finite and algebraic extensions, degree of extension, splitting field of a polynomial. Algebraically closed fields, the fundamental theorem of algebra, multiple roots and derivative criterion, classification of finite fields, Frobenius morphism, n-th roots of unity and cyclotomic extensions. Constructions with row and compass (the problems of the trisection of the angle, the quadrature of the circle, the rectification of the circumference and the duplication of the cube). A brief introduction to Galois theory in 0 characteristic.

It useful for the student to know and master the basics imparted in the first courses of linear algebra.

Israel Herstein, Algebra, Editori Riuniti Michael Artin, Algebra, Bollati Boringhieri S. Weintraub, Galois Theory Lecture Notes on Groups Theory by J. Milne

Attending is not mandatory.

The exam aims to evaluate learning through a written test (consisting in solving problems of the same type as those carried out in the exercises) and an oral test (consisting in the discussion of the most relevant topics illustrated in the course). The written test will last about three hours and can be replaced by two intermediate tests, both lasting at least two hours, the first of which will take place in the middle and the second at the end of the course. The first intermediate test will focus mainly on the topics of arithmetic and group theory, the second on the remaining topics of the course. Students who have obtained a grade of not less than 17/30 in the written test (or the average of the two intermediate tests) are admitted to the oral exam. The minimum grade to pass the exam is 18/30. The student must demonstrate that he has acquired sufficient knowledge of the topics of all parts of the program. To achieve a score of 30/30 cum laude, the student must demonstrate that he has acquired excellent knowledge of all the topics covered during the course and be able to link them in a logical and coherent manner.

The course consists of lectures and exercise sessions.