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 hassynthesis 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. Polynomials; zeros and linear factors; K[X] is an euclidean domain; consequences. 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 uniquenessof 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

Teaching tools

Additional excercise sheets can be found athttp://www.dm.unibo.it/~ida/annoincorso.html

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

Links to further information

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

Office hours

See the website of Monica Idà