54772 - Algebra II (First Part)

Learning outcomes

The aim of the course is to continue the study of groups, rings (in particular the study of the polynomials ring in one variable) and fields, started in the course of Algebra I

Course contents

I – HOMOMORPHISMS OF GROUPS AND QUOTIENT GROUPS. Homomorphisms. of groups. Kernel and image. Isomorphisms. Conjugation. Characterization of conjugate elements in symmetric groups. Normal subgroups and quotient groups. Classification of groups of order less or equal 7. Fundamental Theorem on homomorphism of groups. Homomorphisms from Z to a group. Classification of cyclic groups. Quotient of permutation group on 4 letters with respect to Klein’s group II - HOMOMORPHISMS OF RINGS, IDEALS AND QUOTIENT RINGS Homomorphisms of commutative rings. Homomorphisms of rings from Z to a commutative ring. Kernel and image of homornorphisms of commutative rings. Homomorphisms from a field to a commutative ring. Isomorphisms of commutative rings. Ideals in a commutative ring. Quotient rings and their ideals. Characterization of fields as rings without no trivial ideals. Prime ideals, maximal ideals and their characterizations. Ideals in polynomials rings with coefficients in a field. Fundamental Theorem on homomorphism for rings. Chinese remainder theorem. Bijective correspondence between the ideals in a ring A containing an ideal I and the ideals in the quotient ring A/I. Characteristic of a ring. Characteristic of an integral domain. Order of a finite field as a power of the characteristic. Quotient field of an integral domain. Quotients field as the smallest field containing a given integral domain. Every field with characteristic zero (resp. with characteristic p>0) contains a subfield isomprphic to Q (resp. to Z/p).

Readings/Bibliography

A.Vistoli: Lezioni di Algebra. Bologna, 1993-94 A.Conte - L.Picco Botta - D.Romagnoli: Algebra Levrotto e Bella, Torino 1990 I.N.Herstein: Algebra. Editori Riuniti, Roma 1994 E. Bedocchi: Esercizi di Algebra. Pitagora Editrice Bologna, 1995-96.   Books containing further exercises: A.Alzati - M.Bianchi: Esercizi di Algebra per Scienze dell Informazione. Città Studi, Milano 1991. A.Facchini: Sussidiario di Algebra e Matematica Discreta Decibel - Zanichelli, Bologna 1992 M.Fontana - S.Gabelli: Esercizi di Algebra Aracne Editrice, Roma, 1993 S.Franciosi - F.De Giovanni: Esercizi di Algebra. Aracne Editrice, Roma 1993. R. Procesi Ciampi-R.Rota: Algebra moderna. Esercizi. Editoriale Veschi. Masson, Milano 1992. A.Rugusa - C.Sparacino: Esercizi di Algebra. Zanichelli Editore, Bologna 1992.

Teaching methods

Lectures – exercise sessions – office hours

Assessment methods

Written and oral exam

Teaching tools

Lectures with exercise sessions. Sheets of exercises will be handed out during the lectures, in addition to the ones available in the suggested textbooks. The exercises proposed in these sheets will be solved during exercise sessions. Further material for prepartion to the written exam can be found in all the exercises books of algebra that can be consulted in the library and listed in the bibliography. In the office hours students will be coached individually.

Links to further information

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

Office hours

See the website of Mirella Manaresi