I need some help with clearing up some some basic concepts in finite fields. I understand that $\mathbb{F}_p = GF(p)$ where $p$ is a prime is a finite field, which is isomorphic to $\mathbb{Z}/p\mathbb{Z}$. However, I get quite confused with $GF(p^n)$.
Since $GF(p)$ is a finite field, is $GF(p^n)$ a vector space of dimension $n$ over $GF(p)$ ($n$-tuples of elements from $GF(p)$) or another finite field with more ($p^n$) elements? Can I understand this intuitively as like when $\mathbb{R}$ is the field then $\mathbb{R}^n$ is an $n$-dimensional vector space over $\mathbb{R}$?
Another thing is the $\mathbb{F}_p^n $. Is this the same as $GF(p^n)$? But what's $\mathbb F_{p^n}$ then? Are they just two different notations of the same thing? It seems more logical to me that the first one means something like $GF(p)^n$ but is it something meaningful?
Finally, when someone states "Let $V$ be an $n$-dimensional vector space over a finite field..." - does he explicitly mean $GF(p^n)$ for some $p$ prime or something else?