Algebraic Structures, 7.5 Credits

Contents

The course covers the basic theory of groups rings and field, including concepts like residue classes, ideals, and isomorphisms. Applications of the underlying theory is given in combinatorics, cryptology, coding theory, and geometric constructions.
 
Further, polynomials with coefficients in a field are studied, and how zeros of a polynomial can be found in a larger field. The general theory for such field extensions is then connected to three classical geometric problems; trisection of an angle, doubling the cube, and squaring a circle, and why these are unsolvable.
 

Expected learning outcomes

For a passing grade, the student must be able to
 
Knowledge and understanding

• account for definitions and basic properties of groups, ring, and fields.
• account for the basic theory of polynomials and field extensions

Skills and abilities

• use structure theorems to describe finite groups
• use symmetry groups and groups actions on sets
• factorize polynomials over finite fields
• apply algebraic codes
• translate geometric construction problems to number field problems

Judgment and approach

• decide the applicability of symmetries of mathematical objects.

Required Knowledge

The course requires 60 ECTS in mathematics or at least two years university studies and in both cases courses in Discrete Mathematics and Linear Algebra or equivalent. Proficiency in English equivalent to Swedish upper secondary course English 5/A. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.
 

Form of instruction

The teaching takes the form of lectures.

Examination modes

The course is examined by written exam. On the written exam and for the whole course, one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG). In order to receive a passing grade on the course, all parts must be completed with a passing grade.

A student who has been awarded a passing grade for the course cannot be reassessed for a higher grade. Students who do not pass a test or examination on the original date are given another date to retake the examination. A student who has sat two examinations for a course or a part of a course, without passing either examination, has the right to have another examiner appointed, provided there are no specific reasons for not doing so (Chapter 6, Section 22, HEO). The request for a new examiner is made to the Head of the Department of Mathematics and Mathematical Statistics. Examinations based on this course syllabus are guaranteed to be offered for two years after the date of the student's first registration for the course.

Credit transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications

Other regulations

In a degree, this course may not be included together with another course with a similar content. If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics.