equality between the index between field with $p^{n}$ elements and $ \mathbb{F}_{p}$ and n? can someone explain this?
$ \left[\mathbb{F}_{p^{n}}:\mathbb{F}_{p}\right]=n $

$$|\Bbb F_p|=p\;,\;\;|\Bbb F_n^n|=p^n$$
amd since any element in the latter is a unique linear combination of some elements of it and scalars from the former, it must be that those some elements are exactly $\,n\,$ in number.
