ALGEBRA 2
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Bibliography
- Teaching methods
- Contacts/Info
Algebra 1 course content or equivalent: set and group theory. Some basic linear algebra; vector spaces, matrices, basis, matrix representation of homomorphisms, eigenvalues and eigenvectors.
Written and oral exam.
The written examination lasts 2 hours and 30 minutes and tipically consists of 4 or 5 exercises divided in subquestions. It is required the ability to establish properties of the algebraic structures, to determine the canonical forms of a square matrix and to justify all the statement given.
The written examination is graded on a scale from 0 to 30. Students need a grade of 14/30 or more to be admitted to the oral examination.
The oral examination usually begins with the discussion of the written test. The student will then be required to present some of the results seen in class. It will be object of evaluation the ability to present a proof in a complete and rigorous.
Passing examination and the final grading depend both on oral and written tests. More detail on course web page.
Knowledge of main results in ring and module theory. Applications to the study of canonical form of matrices and to the classification of finitely generated abelian groups.
EXPECTED LEARNING OUTCOMES
At the end of the course the student will be able to establish properties of algebraic structures with more than one operation, to apply Sylow theory to get information about the structure of finite groups, to recognize finitely generated abelian groups and to determine several canonical forms of a square matrix.
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. Theorem di Cayley-Hamilton.
Topics in groups theory
Subgroups of finite p-groups. Sylow's theorem. Semidirect product of groups.
Rings 30 hours
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. Theorem of Cayley-Hamilton.
P.M. Cohn, Classic Algebra, Wiley.
Exercises, exam problems, and lecture notes available on the course web page.
Frontal lectures. Attending lectures is not mandatory, but strongly recommended.
Lectures are given at the board. Every topic is explained together with exercises useful to understand and apply exposed results. Sometimes the solution is given immediately, sometimes in a subsequent lecture in order to stimulate student to autonomous work. Especially at the end of the course, lectures of summarizing exercises are scheduled, in order to choose the more appropriate method to solve an exercise and to establish connections between different topics.
The exercises presented are often taken from past written examinations: they can be found, together with other selected exercise, on the web page of the course.
For simple and short questions, ask the teacher immediately before or after the class. Email teacher for longer questions.
For further detail go to the web page of the course.