Entry requirements

Algebra I.

Learning outcomes

On completion of the course, the student should be able to:

• give an account of important concepts and definitions in the theory of rings and fields;
• exemplify and interpret important concepts in specific cases;
• formulate important results and theorems covered by the course;
• describe the main features of the proofs of important theorems;
• use the theory, methods and techniques of the course to solve simple number theoretic problems and problems about rings and fields;
• present mathematical arguments to others.

Content

Number theory: Congruences, Euler's phi-function, Fermat's little theorem, linear congruences, the Chinese remainder theorem, the RSA algorithm.

An introduction to ring and field theory: Properties of addition and multiplication in Z, Q, R, Z[x] and C[x]. The ring and field concepts. Invertible elements and prime elements. Unique factorisation in Z and in K[x]. The Euclidean ring notion, unique factorisation and the ring Z[i] of Gaussian integers. Isomorphism, homomorphism, ideal, quotient field. The ring Z_n of integers modulo n. Examples of non-commutative rings.

Instruction

Lectures and problem solving sessions.

Assessment

Written (4 Credit Points) and oral (1 Credit points) examination.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.