This course will cover basics of abstract rings and fields, which are an important part of any abstract algebra course sequence. We will spend roughly the 4-5 weeks on rings. We will begin with definitions and important examples. We will focus cover prime, maximal ideals and important classes of rings like integral domains, UFDs and PIDs. We will also prove the Hilbert basis theorem about noetherian rings. The last 3-4 weeks will be devoted to field theory. We will give definitions, basic examples. Then we discuss extension of fields, adjoining roots, and prove the primitive element theorem. Finally we will classify finite fields.INTENDED AUDIENCE: B.Sc and M.Sc students studying mathematicsPREREQUISITES: A little bit of abstract group theory and a little bit of linear algebra.INDUSTRY SUPPORT: None
Week 1: Definition of rings, examples, polynomial rings, homomorphisms.Week 2: Ideals, prime and maximal ideals, quotient rings.Week 3: Noetherian rings, Hilbert basis theorem.Week 4: Integral domains, quotient fields.Week 5: Unique factorization domains, principal ideal domains.Week 6: Definition of fields, examples, degree of field extensions.Week 7: Adjoining roots, primitive element theorem.Week 8: Finite fields.