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:
- GF stands for Galois Field, which means its a finite field
- $[GF(p^n):GF(p)]$ 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?
- 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?
$\endgroup$ 102 Answers
$\begingroup$$[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 $n$
$\endgroup$ 4 $\begingroup$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 $n$.
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