Abstract Algebra: Theory and Applications

Suppose $p(x)$ is a polynomial of degree $n$ with coefficients from any field. How many roots can $p(x)$ have? How does this generalize your high school algebra experience?

What is the definition of an irreducible polynomial?

Find the remainder upon division of $8x^5-18x^4+20x^3-25x^2+20$ by $4x^2-x-2\text{.}$

A single theorem in this chapter connects many of the ideas of this chapter to many of the ideas of the previous chapter. State a paraphrased version of this theorem.

Early in this chapter, we say, “We can prove many results for polynomial rings that are similar to the theorems we proved for the integers.” Write a short essay (or a very long paragraph) justifying this assertion.