Understanding the formula:$ [GF(p^n):GF(p)]=n$

Question

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?

Answer 1

$[GF(p^n):GF(p)]=n$ denote the degree of the extension field $GF(p^n)$ over $GF(p)$ not order of $GF(p^n)$ over order of $GF(p)$

Degree means the dimension of $GF(p^n)$ over $GF(p)$. Here the dimension is $n$

Answer 2

This means you have a finite field with the number of elements (=order) $p^n$, this field can be turned into a linear (=vector) space. In this vector space the scalars are elements of the subfield of this field $GF(p^n)$, prime subfield of order $p$.

The dimension of this vector space is $n$.