Abstract Algebra Lectures Part 18: Polynomial Rings

For a commutative ring, we define R[t] as the set of polynomials in t with coefficients in R. If R is an integral domain, so is R[t].

In this video, we study the evaluation map given by an element in some ring which contains R. This map allows us to define transcendental and algebraic elements, something of importance in Galois theory.

We also look at K[t] for a field K, and show how we can factor out linear terms corresponding to roots, showing that a degree n polynomial has at most n roots.