Course syllabus - Abstract Algebra

Objectives

Algebra is one of the fundamental branches of modern mathematics. It has its origins in the theory of numbers and geometry. The aim of the course is to find, through examples, the mathematical structures underlying concepts in number theory and geometry. These structures, groups, rings and fields, are applied in multiple contexts such as counting and enumeration problems, coding theory and combinatorial designs.

Learning outcomes

After completing the course the student should be able to

• using set theoretical language define and give examples of the basic structures and fundamental concepts of algebraic theory, and appropriately use the formal language of the theory in speech and writing
• formulate, interpret and give examples of the basic facts and constructions of the theory
• using formal reasoning prove or disprove the simple statements in the theory
• place the theory in its context in mathematical history and give examples of the connection between algebraic theory and other branches of Mathematics such as geometry or analysis

Course content

• Sets, equivalence relations
• Groups: subgroups, permutation groups, cyclic groups, cosets, direct product, Abelian groups, honomorphisms, quotient groups, simple groups
• Rings and fields: integral domains, ideals, homomorphims and quotient rings. Maximal ideals, polynomial rings, factorization, field of quotients of an integral domain
• Field extensions: algebraic extensions, constructibility. Finite fields.
• Coding theory

Specific requirements

120 credit points in Engineering, Natural Science, Business Administration, or Economics, including at least two of the courses Vector Algebra 7.5 credit points, Discrete Mathematics 7.5 credit points, Linear Algebra 7.5 credit points, or the equivalent. In addition, Swedish course 3 or Swedish level 3 and English course 6 or English level 2 are required. For courses given entirely in English exemption is made from the requirement in Swedish course 3 or Swedish level 3.

Examination

Final exam, written and/or oral (TEN1), 7.5 credits, marks 3, 4 or 5, Final exam, written and/or oral. May be partially or fully replaced by written assignments.

A student who has a certificate from MDU regarding disability study support, can request adaptions for the examination. It is the examiner who takes decisions on any adaptions, based on the certificate and other conditions.

Grade

Grading scale: 5, 4, 3