MM103: Algebra I
Study Board of Science
Teaching language: Danish
EKA: N900003102
Assessment: Second examiner: External
Grading: 7-point grading scale
Offered in: Odense
Offered in: Autumn
Level: Professional Master
STADS ID (UVA): N900003101
ECTS value: 5
Date of Approval: 25-04-2019
Duration: 1 semester
Version: Archive
Comment
Entry requirements
Academic preconditions
Students taking the course are expected to:
- Have knowledge of the basic numbersystems and associated compositions
- Be able to use and perform elementary mathematical reasoning and calculations.
Course introduction
In abstract algebra the fundamental objects of study are sets with one or more compositions. A composition is a mapping that associates to any two elements of the set a third element. For example the set of integers and the set of polynomials both have two compositions: addition and multiplication. Sets with compositions appear everywhere in mathematics. The algebraic theory for such sets has many applications both within pure mathematics and in applications of mathematics. The goal of the course is to make the students feel acquainted with the basic algebraic concepts through important examples like the integers and polynomials, and with applications in e.g. cryptology. The participants will learn how an abstract theory can be constructed based on axioms, and how it can be applied in concrete cases.
The course builds on the knowledge acquired in courses in calculus and linear algebra, which are mandatory prerequisites for students entering the masters studies in mathematics with focus on teaching in high school educations.
The course gives an academic basis for studying e.g. the topics geometry and algebra 2 that are part of the degree.
In relation to the competence profile of the degree it is the explicit focus of the course to:
- Give the competence to teach topics related to algebra in high school educations at a highly qualified level.
- Give skills in applying the algebraic techniques, results and concepts on concrete examples in e.g. number theory, rings and polynomials.
- Give knowledge and understanding of the basic structure of the number systems and their compositions.
Expected learning outcome
The learning objective of the course is that the student demonstrates the ability to:
- Apply algebraic algorithms on concrete problems. Examples could be Euclids algorithm and division of polynomials.
- Argue for the various steps in algebraic problems with reference to the relevant theory.
- Perform calculations modulo an integer and in rings of polynomials over a field.
Content
The following main topics are contained in the course:
- The basic number systems
- Elements of group theory, rings and polynomials.
- Primenumbers, primefactorization of integers, cryptology, quadratic remainders.
- Introductory group theory, rings, polynomials and finite fields.
Literature
Examination regulations
Exam element a)
Timing
January
Tests
Written exam
EKA
N900003102
Assessment
Second examiner: External
Grading
7-point grading scale
Identification
Student Identification Card
Language
Normally, the same as teaching language
Examination aids
A closer description of the exam rules will be posted under 'Course Information' on Blackboard.
ECTS value
5
Additional information
The examination form for re-examination may be different from the exam form at the regular exam.
Indicative number of lessons
Teaching Method
At the faculty of science, teaching is organized after the three-phase model ie. intro, training and study phase.