Algebraic Structures, 7.5 Credits

Contents

In the course, classical abstract algebra is studied, where concepts such as groups, rings, integral domains and fields, as well as  residue classes, ideals, and isomorphisms are central. Applications of the fundamental theory of these structures is given in combinatorics, cryptology and coding theory.
 
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 the 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

• analyse finite groups using structure theorems
• perform proofs of theorems in abstract algebra
• apply abstract algebra in combinatorics, coding theory and questions regarding geometric constructions

Judgment and approach

• place algebraic concepts in a wider context with regards to history, other areas of mathematics, or applications

Required Knowledge

The course requires 60 ECTS in mathematics and mathematical statistics at least two years university studies and in both cases courses in Discrete Mathematics and Linear Algebra or equivalent. Proficiency in English equivalent to the level required for basic eligibility for higher studies. 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 hand-in exercises, oral presentations and a written exam. For hand-in exercises and the written exam one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG). For oral presentations, one of the following grades is assigned: Fail (U), Pass (G). In order to receive a passing grade on the course, all parts must be completed with a passing grade. To receive the grade Pass with distinction (VG) for the whole course, this grade must be acchieved on both the hand-in exercises and the written exam.

Deviations from the syllabus examination form can be made for a student who has a decision on pedagogical support due to disability. Individual adaptation of the examination form shall be considered based on the student's needs. The examination form is adapted within the framework of the expected learning outcomes of the course syllabus. At the request of the student, the course coordinator, in consultation with the examiner, must promptly decide on the adapted examination form. The decision shall then be communicated to the student.

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 similar contents. If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics.