I came across the following in my Cryptography notes:
Definition: Let $p$ denote a prime number, and let $F_p$ denote the field with $p$ elements. Let $f(x)$ denote an irreducible polynomial over $F_p$ of degree $d$. The finite field $F_{p^d}$ with $p^d$ elements, is simply the quotient ring:
$F_{p^d} = F_p[x]/(f) = \{ \sum_{i=0}^{d-1}a_ix^i ; a_i \in F_p$ and $f=0 \}$. The polynomial $f(x)$ is called the defining polynomial of the finite field $F_{p^d}$.
From my understand of ring theory, $F_p[x]/(f) = \{p(x) + (f)|p(x) \in F_p[x] \}$, so I am not sure how this definition was arrived at. Any insights appreciated.