Field Extensions and Roots of Polynomials (Algebra 3: Lecture 8 Video 2)

Lecture 8: We started this lecture by giving a proof of Theorem 4 in Section 13.1. We pointed out the close relationship between this result and a general fact about bases of quotients of the form F[x]/(f(x)) that we proved in Math 206B. We saw how this result gave us a concrete description of elements of K = F[x]/(p(x)) and a nice way to add and multiply them. We also discussed how to find inverses of elements in this field. We discussed several examples. For example, we showed that R[x]/(x^2+1) is isomorphic to C. We also constructed a finite field of order 4 and wrote down the addition and multiplication tables for it. We discussed some questions about finite fields that we will return to later in the course. We then defined the field generated by a set of elements over F and what it means for K/F to be a simple extension. We stated the Primitive Element Theorem, which we will prove later in the course. We proved a result relating the fields F(alpha) and F[x]/(p(x)) and discussed the examples F = Q and p(x) = x^2 -2 and F = Q and p(x) = x^3-2.

Reading: We closely followed Section 13.1 from Theorem 4 on page 513 to the bottom of page 518.  We also mentioned some results that we will see later in the course-- a result about finite fields (part of Proposition 15 in Section 14.3), the Primitive Element Theorem (Theorem 25 in Section 14.4), and the concept of what it means for a field extension to be separable (Section 13.5).  You can look ahead at these part of the textbook to get a sense for what is coming up, but it is fine if you wait.