Polynomial Rings and Matrix Rings (Algebra 1: Lecture 26 Video 5)

Lecture 26: In this lecture we continued our discussion of examples of rings. We started by talking about the (real) Hamilton quaternions and some related rings of quaternions. These were nice examples of non-commutative rings. We then extensively discussed Quadratic Fields and Quadratic Rings. We discussed the norm function in a quadratic field and saw how it behaved as a function on the ring of integers of a quadratic field. We proved that an element in such a ring is a unit if and only if it has norm 1 or -1. We saw how this determined the units in O_{Q(sqrt{D}) when D is a negative squarefree integer. We then briefly introduced two important classes of rings; Polynomial Rings and Matrix Rings.

Reading: In this lecture we closely followed the end of Section 7.1 of Dummit and Foote, and then most of Section 7.2. We skipped the discussion of Group Rings at the end of Section 7.2-- I do not plan on discussing Group Rings in this course.