Course syllabus - Introduction to Algebraic Structures

Objectives

The course aims at giving the student a basic knowledge, tools and methods on the topic of algebraic structures.

Learning outcomes

At the end of a passed course, the student is expected to be able to

- define and give examples of those algebraic structures that are covered in the course
- find all rational and multiple zeroes of a polynomial with rational coefficients
- reduce to normal form any rational expression of a root of an irreducible polynomial
- prove the Fundamental Theorem of Algebra and the lemmata on which that theorem is based
- carry out simple proofs based on the concepts treated in the course
- explain, evaluate, and analyse arguments employing the concepts treated in the course

Course content

Polynomials: zeroes, the Factor Theorem, greatest common divisor, Euclid’s Algorithm, criteria for irreducibility, the formal derivative, the Fundamental Theorem of Algebra, algebraic number fields.

Algebraic structures: groups and semigroups, rings, integral domains, fields, vector spaces over general fields. Composition of functions as the operation of a group or semigroup. Permutations. Matrix rings and groups. The ring of integers, residue class rings.

Tuition

Teaching is given in the form of lectures and classes.

Specific requirements

At least totally 60 credits in the technical, natural sciences, business administration or economics areas including Linear Algebra, 7.5 credits, of which 5 credits must be completed at the beginning of the course, or the equivalent.

Examination

Assigned problems (INL1), 2,5 credits, marks Fail (U), Pass (G)
Written and/or oral examination  (TEN1), 5 credits, marks Fail (U), 3, 4, 5

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

Grading scale: 5, 4, 3