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

At least 120 credits in the technical, natural sciences, business administration or economics areas where Linear Algebra 7,5 credits or equivalent is included. In addition Swedish course B/Swedish course 2 and English course A/English course 5 are required. For courses given entirely in English exemption is made from the requirement in Swedish course B/Swedish course 2.

Examination

Final exam, written and/or oral. May be partially or fully replaced by written assignments (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 a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Pass with distinction, Pass with credit, Pass, Fail