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 $q(x)$
and
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?
These are exactly the same (note though that this field is written $\mathbb{F}_{p^n}$, not $\mathbb{F}_p^n$). "Residue classes mod $q(x)$" and "elements of the quotient by the ideal generated by $q(x)$" are two different terms for the same thing: namely, equivalence classes of elements of $\mathbb{F}_p[x]$ under the equivalence relation $f(x)\sim g(x)$ iff $f(x)-g(x)$ is a multiple of $q(x)$. The difference in notation between $(q(x))$ and $\langle q(x)\rangle$ is meaningless--some people use parentheses to denote ideals generated by a list of elements, and other people use angle brackets. You should feel free to use either one, as long as you are consistent.