$F$ is a field. $\langle p(x) \rangle$ is a maximal ideal. So $K = F[x]/\langle p(x) \rangle$ is a field extension.
I am trying to understand what would be the structure of $K[x]$?
$F[x]$ has all polynomials where coefficients of the polynomials are from the Field $F$. So $K[x]$ would be the set of polynomials with their coefficients are from the field $K$.
Every element of $K$ is a coset of the form $a(x) + \langle p(x)\rangle$ where $a(x) \in F[x]$ and degree of $a(x)$ is lesser than that of $p(x)$ and the coset can be denoted as $[a(x)]$. I am not able to understand how a coset can be a coefficient of a polynomial. I think probably the elements of the cosets could be. The element of each coset is apart of the equivalence class $[a]$. So I think the coefficients of the polynomials in $K[x]$ would be the different $a$'s. If that is true, then I think the set of polynomials in $K$ would be exactly the same as the set of polynomials in $F$ - i.e. $K[x]$ would be the same as $F[x]$. Or am I mistaken?