Hey everyone, I was hoping to grab some quick advice on these two topics. Specifically, I'm a 4th year physics undergrad with all the standard physics and math courses, as well as real analysis up to lebesgue measure theory/integration theory+hilbert spaces,etc., and grad level PDEs. I have plans to take some abstract algebra courses as well as topology and a couple of other things this coming semester. Specifically for abstract algebra, I'm looking into ring theory and group theory. I understand group theory is used more in physics, however ring theory appears to be a prerequisite to the course. In the past this was never the case and you could take one without the other. My question is, with the follow course descriptions, how much of group theory will I not be understanding without the ring theory course? Just want to get some outside advice on the subject before talking to the profs. Ring theory: Integers. Mathematical induction. Equivalence relations. Commutative rings, including the integers mod n, complex numbers and polynomials. The Chinese remainder theorem. Fields and integral domains. Euclidean domains, principal ideal domains and unique factorization. Quotient rings and homomorphisms. Construction of finite fields. Applications such as public domain encryption, Latin squares and designs, polynomial error detecting codes, and/or addition and multiplication of large integers. Group theory: Groups as a measure of symmetry. Groups of rigid motions. Frieze groups, and finite groups in 2 and 3 dimensions. Groups of matrices. Group actions with application to counting problems. Permutation groups. Subgroups, cosets, and Lagrange's Theorem. Quotient groups and homomorphisms. Coto
i guess there´s pretty few group therory stuff you won´t be able to gasp not knowing ring therory. Maybe it´s a good idea to have some knowledge of the integers mod n but even that isn´t that cruicial in my opinion i don´t really see a problem taking groups without ringtheory, while ringtheory is in some sense an extension of grouptheory :)
Ring Theory a prerequisite to Group Theory?! Usually it's the other way around. Anyways, neither topic rely on each other at the introductory level. It doesn't matter which you take first.