Fields of the form k[x]/I where I = (p(x)) I was reading Advanced Modern Algebra by Rotman and I came across something that confused me.

On page 186 Proposition 3.117 (iii), he mentioned that $k[x] \subseteq K[x]$ where $K = k[x]/I$ with $I = (p(x))$ and $p(x)$ is a monic irreducible polynomial and $k$ is an arbitrary field. However, $k$ isn't even a subset of $k[x]/I$. Why is this the case?
The canonical projection $k[x] \to k[x]/I$ is injective on $k$ and so there is an isomorphic copy of $k$ in $k[x]/I$.
There is a canonical (and injective) homomorphism $k[x]\to K[y]$ (different variable names for clarity) sending $k\ni 1\mapsto 1+I\in K$ and $x\mapsto y$.