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School of Science
Department of Mathematics
|Level of Studies||
Algebraic Structures I
|Independent Teaching Activities||
Lectures (Weekly Teaching Hours: 5, Credits: 7.5)
|Language of Instruction and Examinations||
|Is the Course Offered to Erasmus Students||
Yes (in English)
|Course Website (URL)||See eCourse, the Learning Management System maintained by the University of Ioannina.|
The course aims to introduce the students to the study algebraic properties of sets which are equipped with one or more (binary) operations. Such mathematical objects are called algebraic structures. We will mainly deal with two types of algebraic structures:
We will formulate various theorems concerning the structure and basic properties of groups and rings emphasizing the concept of isomorphism of groups or rings. From the perspective of Algebra two algebraic structures which are isomorphic, they have exactly the same algebraic properties. As a direct consequence, results concerning an algebraic structure are valid in any isomorphic algebraic structure. In the course we present several examples illuminating various notions of symmetry. It should be noted that the notion of symmetry is the central theme which underlies the concept of group/ring.
The course aims to enable the undergraduate student to acquire the ability to analyse and synthesize basic knowledge of the theory of algebraic structures, in particular of the general theory of Groups and Rings, which form an important part of modern algebra. The contact of the undergraduate student with the ideas and concepts of the theory of groups and rings, (a) promotes the creative, analytical and deductive thinking and the ability to work independently, (b) improves his critical thinking and his ability to apply abstract knowledge in various field.
- Preliminaries: Sets, functions, equivalence relations, partitions, (binary) operations.
- Groups – Permutation groups.
- Cyclic groups – generators.
- Cosets with respect to a subgroup – Lagrange’s Theorem.
- Homomorphisms of groups – Quotient groups.
- Rings and fields - Integral domains.
- The theorems of Fermat and Euler.
- Polynomial rings – Homomorphisms of Rings.
- Quotient rings – Prime and maximal ideals.
Teaching and Learning Methods - Evaluation
Classroom (face to face)
|Use of Information and Communications Technology||
Teaching Material: Teaching material in electronic form available at the home page of the course.
|Student Performance Evaluation||
Final written exam in Greek (in case of Erasmus students, in English) which includes analysis of theoretical topics and resolving application problems.