Degree of a Finite Field

Question

Consider the finite field of characteristic $\mathbb{F}_{p}$ and the polynomial $f(x) = x^{p^{n}}$ - x. The splitting field $f(x)$ is a field $\mathbb{F}_{p^{n}}$ with $p^{n}$ elements. Given this information could somebody enlight me as why $[\mathbb{F}_{p^{n}} :\mathbb{F}_{p}] = n$?

Thanks

Answer 1

If E is any vector space over $F_p$ that is $n$ dimensional, the E has $p^n$ elements. This is just counting n tuples over the finite field.

Since $F_{p^n}$ has $ p^n$ elements, there is simply no other possibility for the degree of the extension.