You are here:

28368 - Algebra 2

Academic Year 2021/2022

                
                        Docente:
                        Marilena Barnabei
                    
                        Credits:
                        7
                    
                        SSD:
                        MAT/02
                    
                        Language:
                        Italian
                    
                        Teaching Mode:
                        Traditional lectures
                        
                            Campus:
                            Bologna
                        
                            Corso:
                            First cycle degree programme (L) in
                            Mathematics (cod. 8010)

                            Teaching resources on Virtuale

Course contents

Rings: definition and first properties. Special rings: integral domains, fields. Ideal. Ring homomorphisms and isomorphisms. Rings of integers modulo n. Matrix rings. Quaternions.

Divisibility and factorization. Irreducible and prime elements. Unique factorization domains. Greatest common divisor.

Maximal ideals. Prime ideals. Principal ideal domains. Euclidean domains. Gauss integers.

Polynomial rings: definitions. Polynomial long division. Roots and factorization. Polynomial factorization in Z[x] and Q[x].

characteristic of a ring [https://en.wikipedia.org/wiki/Ring_(mathematics)] . Fundamental subring. Frobenius homomorphism. Fermat’s little theorem.

Quotient rings. Homomorphism theorems. Field of fractions of an integral domain. Quotients of a PID and of F[x].

Field extensions: simple extensions. Algbraic extensions. Degree of an extension. Splitting fields.

Finite fields.

Readings/Bibliography

M.Artin: Algebra. Bollati Boringhieri 1997.

I.N. Herstein: Algebra. Editori riuniti, 2010.

Teaching methods

Lectures (48 hours) and exercise sessions (12 hours).

Assessment methods

Written and oral exam

Office hours

See the website of Marilena Barnabei