Basic concepts in finite fields

Question

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?

Answer 1

If $q = p^n$ for some prime $p$, then the notations $\rm{GF}(q)$ and $\mathbb{F}_q$ mean the same, namely the (unique up to isomorphism) field with $q$ elements.

This is indeed a vector space over $\rm{GF}(p)$ of dimension $n$.

$\rm{GF}(p)^n$ and $\mathbb{F}_p^n$ also mean the same, namely the set of $n$-tuples of elements from the field $\rm{GF}(p)$, which is also a vector space of dimension $n$ over $\rm{GF}(p)$.

Since these two vector spaces have the same dimension, they are isomorphic as vector spaces. However, they are not isomorphic as rings, since the first is a field, and the other has non-trivial zero-divisors (take for example en element like $(1,0)$ in $\rm{GF}(p)^2$).

When someone says "a vector space of dimension $n$ over $\rm{GF}(p)$" he means just that. You can take any of the two above and it will not matter, since as vector spaces, we cannot distinguish them.