so as mentioned in the title, I want to understand the formula : $[GF(p^n):GF(p)]=n$ 

So here is what I think it means: 

1. GF stands for Galois Field, which means its a finite field
2. $[GF(p^n):GF(p)]$ can be read as $|GF(p^n)|/|GF(p)|$. So its a division of the orders of both finite fields. The order is in between the brackets, so basically its $(p^n)/p$, but this is not the given $n$ as the result. What is wrong here?
3. Now my biggest concern was to what is the reason this formula is convenient? My answer: The closest thing I could connect this formula to is cosets, where $|G|/|H|$ (H subgroup of G) is the number of cosets of the subgroup. So in this case $n$ is the number of cosets of GF(p) in $GF(p^n)$?

Am I going in the right direction here? Am I missing things?