Rings and Fields

COURSE OBJECTIVE
* The student knows basic concepts of ring theory (ring, homomorphism, ideal, integral domain, field, units, zero divisors, PID, Euclidean
domain, UFD) and can solve problems about and with those in explicit situations.
* The student knows quotient rings, prime/maximal ideals, and theorems relating to those (1st isomorphism theorem, Chinese remainder theorem, recognizing those types of ideals from quotient rings) and can apply those in explicit situations.
* The student can determine a factorization in certain UFDs using some irreducibility tests.
* The student knows some elementary field theory (algebraic xtensions, degrees of extensions, splitting fields, finite fields) and can apply it
in explicit situations.

COURSE CONTENT
This course studies an important algebraic structure (called a ring), which has an addition and a multiplication satisfying certain
properties. Rings arise in many situations, and examples include the integers, the integers modulo n, matrix rings, and polynomial rings, but also the set of complex numbers with integral real and imaginary parts. As is common in algebra, by formalising the common properties we can perform general constructions and prove general results that apply in many contexts, and we illustrate these by working out what they say in various concrete cases. We also study particular types of rings with properties similar to those of the integers (division with remainder, unique factorisation). We conclude by constructing finite fields and some of their properties. These finite fields are used frequently in combinatorics, and they are essential to the theory of error correcting codes (in, for example, QR-codes or electronic train tickets).

TEACHING METHODS
Lectures (14x2 hours) and tutorials (14x2 hours)

TYPE OF ASSESSMENT
Two partial exams (or a resit), and marked assignments. The assignments in total count for 5% towards the grade, the average of scores for the two partial exams (or the score of the resit) counts for 95%. Alternatively, the result of the resit counts 100% if this results in a
higher score.

RECOMMENDED BACKGROUND KNOWLEDGE
The VU courses Group Theory, and Linear Algebra. Although the precise technical knowledge from those two courses that is needed is limited, the (algebraic and abstract) way of thinking that is developed in them is very important for this course.