https://studiegids.vu.nl/en/courses/2025-2026/X_400630The student knows basic concepts of ring theory (ring, homomorphism, ideal, integral domain, field, units, zero divisors, Euclidean domain, principal ideal domain, unique factorisation domain) and can solve problems about and with those in explicit situations.The student knows quotient rings, prime/maximal ideals, and theorems relating to those (first isomorphism theorem, Chinese remainder theorem, recognising those types of ideals from quotient rings) and can apply those in explicit situations.The student can determine a factorisation in certain unique factorisation domains using some irreducibility tests.The student knows some elementary field theory (algebraic extensions, degrees of extensions, splitting fields, finite fields) and can apply it in explicit situations.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 mean 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). We treat the following topics.The definitions of ring, unit, zero divisor, subring, integral domain, field; examples.Ideals, generators.Quotient rings, isomorphism theorems.The Chinese remainder theorem for rings.Polynomial rings, roots, division with remainder (in one variable).Prime ideals, maximal ideals.Euclidean domains, principal ideal domains, unique factorisation domains.Factorisation of polynomials, Eisenstein's irreducibility criterion.Field extensions, degree of a field extension.Finite fields.Lectures (14x2 hours) and tutorials (14x2 hours)Two partial exams (or a resit), and marked assignments. The assignments in total count for 5% towards the grade, the average of the scores for the two partial exams (or the score of the resit) counts for 95%. Alternatively, the result of the exam or the resit counts for 100% of the grade if this results in a higher score. All of this is subject to the requirement that the average of the scores for the two partial exams (or the score of the resit) is at least 55%, otherwise the grade is capped at 5.David S. Dummit, Richard M. Foote, "Abstract algebra", 3rd edition (2003), John Wiley and Sons.Bachelor Mathematics Year 2The 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.