28368 - Algebra 2

Course Unit Page

Academic Year 2018/2019

Learning outcomes

At the end of the course the student has a basic knowledge of advanced algebra. He knows the fundamental algebraic structures of groups, rings and fields. He can apply these tools to other branches of mathematics.

Course contents

Basic facts about group theory. Properties of cyclic groups and chatecterization in terms of subgroups

Commutative rings: 0-divisors, nilpotent elements, units.  Integral domains, fields. Quadratic extensions. Gaussian integers Euclidean domains: principal ideals, GCD, factorization into irreducible factors.  Ring morphisms; the morphism from Z to a ring; characteristic of a ring. Ideals and quotient rings; the ideal generated by a subset. The factorization of a ring morphism. The field of fractions of an integral domain, Q, K(X). Divisibility in a ring. Euclidean domains. Maximal and prime ideals.

The complex numbers; geometric representation, De Moivre Theorem, the n-roots of a complex number.

The polynomial ring with coefficient in a ring, in a domain, degree. The polynomial ring with coefficient in a field, zeros and linear factors, the euclidean division for polynomials and its consequences (K[X] is an euclidean domain, an UFD, a PID). The fundamental Theorem of Algebra. Derivative of a polynomial; multiple roots. Real polynomials.

Quotients of K[X]; reduced form. Field extensions, algebraic and trascendental elements, minimal polynomial; the subfield K(u) of a field F generated by the subfield K of F and the element u. The degree of a finite extension; each element of a finite extension is algebraic; the Tower Theorem; the field of algebraic numbers is algebraically closed. Splitting fields: existence and uniqueness. Existence and uniqueness of the field with p^n elements; these are the only finite fields.

Readings/Bibliography

I.N. Herstein: Algebra, Editori Riuniti, 2010
A.Vistoli: Note di Algebra. Bologna 1993/94
M.Artin: Algebra. Bollati Boringhieri 1997.
E.Bedocchi: Esercizi di Algebra. Pitagora Editrice, Bologna 1995/96

F. Caselli, note di Algebra 2, available online

Teaching methods

Front lectures with exercise sessions

Assessment methods

Written and oral examination. The written assignment consists in some exercises where the student should be able to use the tools acquired during the lessons.  The written exam is accomplished with a note of at least 15/30 and allows the student to be admitted to the oral examinations, where he/she discusses the written assignment and should be able to reason about arguments related to the course. 

Office hours

See the website of Fabrizio Caselli