Let $F_p[x]$ be a ring of polynomials with coefficients in the field $F_p$. Let $q(x)$ be some irreducible polinomial in the base field $F_p$ with degree $n$. To construct the field extension $F_p^n$, I've seen in literature two different notations/constructions:
1) $F_p^n = F_p[x]/\langle q(x) \rangle$ : quotient ring by the ideal generated by $q(x)$
2) $F_p^n = F_p[x]/( q(x) )$ : the residue class by $q(x)$, when operations are done $\mod q(x)$.
Now I ask: These different notations means the same? Why one might be preferable instead the other?