Entry requirements

Algebra I.

Learning outcomes

In order to pass the course (grade 3) 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..