28368 - Algebra 2

Learning outcomes

At the end of the course the student has got the fundamental knowledges of algebraic structures: groups, rings and fields. He  has seen their application in other mathematical fields. He is able to construct a mathematical formalization of problems coming from applied sciences and practical problems. He has synthesis and analysis skillfulness.

Course contents

Quotient groups: equivalence relations associated to a subgroup, Lagrange Theorem, normal subgroup. Group morphisms, the First Fundamental Theorem, the correspondence between subgroups in a group morphism and the subgroup of a quotient group. 

Rings: 0-divisors, nilpotents, units, integral domains, fields. Subrings. The complex numbers; geometric representation, De Moivre Theorem, the n-roots of a complex number. 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 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

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

www.dm.unibo.it/matematica/algebra.htm

Teaching methods

Lectures and exercise sessions

Assessment methods

Written and oral examination. The written assignment consists in some exercises through which the student should prove to be able to use the instruments acquired during the lessons. The student is allowed to bring and use books and written texts during the written exam. If the written exam is sufficient, the student is admitted to the oral examinations, where he/she discusses the written assignment and has to prove to to know and be able to reason about arguments related to the course.

Teaching tools

Additional excercise sheets and previous written examinations can be found at http://www.dm.unibo.it/~ida/annoincorso.html

Some arguments treated in this course can be found at http://progettomatematica.dm.unibo.it/indiceGenerale5.html .

Links to further information

http://www.dm.unibo.it/~ida/

Office hours

See the website of Monica Idà