Why a field $F$ sits naturally inside the ring $F[x]_{m(x)}$ for $m(x)\in F[x]$?

Question

I am reading exercise page 231 from Ronald S. Irving's book: Integers, Polynomials and Rings. I can understand and solve problem in part 1. I get ring $\mathbb{F}_2[x]_{x^2}$ is not a field. But, I am stuck in solving part 2 and 3. How can we show that field $F$ sits naturally inside the ring $F[x]_{m(x)}$ for $m(x) \in F[x]$. I appreciate any help. Thank you.

Answer 1

There is a natural embedding of of $F$ into $F[x]$. try to show its kernel is exactly $(x)$