Linear independence over an extension

Question

I'm working with the extension $k(p):k(p^n)$

where p is an indeterminate.

I'm trying to show that ${1,p,p^2,...,p^{n-1}}$ is linearly independent on $k(p^n)$

I started by assuming they're linearly dependent and then trying to arrive at a contradiction using the shortest relation of dependence i.e $a_0+a_1p+...+a_mp^m=0$ where all $a_i \neq 0$ and $m\leq n-1$

But haven't been able to progress, any help would be appreciated.

Answer 1

$p$ is a root of the polynomial $f=X^n-p^n$. It suffices to prove that this polynomial is irreducible over $k(p^n)$ since then $[k(p):k(p^n)]=n$. By Gauss's lemma, $f$ is irreducible over $k(p^n)$ if and only if it is such over $k[p^n]$; and this follows from Eisenstein's criterion because $p^n$ is irreducible over $k[p^n]$.

Answer 2

I don't know how to write. But, Let $g(p)=\sum_{i=0}^{n-1}f_i(p^n)p^i$. Clearly $g$ has a non zero leading term, so its never zero