ALGEBRA 2

Degree course: 
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree: 
2015/2016
Year: 
2
Academic year in which the course will be held: 
2016/2017
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
First Semester
Standard lectures hours: 
64
Detail of lecture’s hours: 
Lesson (64 hours)
Requirements: 

Algebra 1 course content or equivalent: set and group theory.

Final Examination: 
Orale
Assessment: 
Voto Finale

Teaching objectives and expected learning outcomes
Knowledge of main results in ring and module theory.

Course program
Topics in groups theory
Subgroups of finite p-groups. Sylow's theorem. Semidirect product of groups.

Rings
Definitions. Zero divisors. Invertible elements. Integral rings and integral domains. Division rings. Fields. Quaternions. Field of fractions of a domain.
Subrings, ring homomorphisms. Isomorphism theorem. Ideal and quotient ring. Characteristic of a ring. Prime subfield of a ring.
Polynomial rings. Formal power series ring. Universal property of polynomial rings. Multivariate polynomials.
Divisibility in integral domain. Associate elements. Prime and irreducible elements. Unique factorization domains (UFD) and their characterization. Highest common factor and least common multiple. Euclidean domains. Euclid's algorithm. Division of polynomials. Zeroes of polynomials. Factorization of polynomials. Gauss' lemma. Theorem: the polynomial ring over a UFD is a UFD.
Extensions of fields. Algebraic elements. Minimal polynomial. Transcendent elements. Roots of unity. Finite fields.

Modules
Endomorphism ring of an abelian group. Left and right modules over a ring. Opposite ring. Left and right ideals. Submodules. Module homomorphisms. Direct product of modules. Independent modules. Linear independent elements. Submodule generated by a subset. Free modules. Basis. Annihilators of elements and modules. Cyclic modules.
Principal ideal domains (PID). Classification of finitely generated modules over PID. Finitely generated abelian groups. Jordan's canonical form of a matrix. Minimum polynomial of a matrix.

Type of didactic activities
Frontal lectures and guided exercises

Texts and teaching material

P.M. Cohn, Classic Algebra, Wiley.

Exercises, exam problems, and lecture notes available on the class web page.

For further detail go to the web page

Verification of learning skills
Written and oral exam.
The written examination lasts 2 hours and 30 minutes and tipically consists of 4 or 5 exercises divided in subquestions.
The written examination is graded on a scale from 1 to 30. Students need a grade of 14/30 or more to be admitted to the oral examination.
Passing examination and the final grading depend both on oral and written tests.

Professors