$\newcommand{\char}{\text{char}}$ Let $K$ be a field and $\char(K)=p\not =0.$ Prove that the polynomial $X^{p^n}-a\in K[X]$ for $a\in K$ has at most one root.
My Attempt: Suppose $m\not = n$ in $K$ are the two roots of the polynomial. Denote $p^n=\lambda$ then $m^\lambda=n^\lambda.$ Thus $$(m-n)(m^{\lambda-1}+nm^{\lambda-2}+...+n^{\lambda-1}).$$ Now $m\not =n$ so $$m^{\lambda-1}+nm^{\lambda-2}+...+n^{\lambda-1}=0.$$ How can I use this to arrive at a contradiction?
You are not using the essential fact that $\lambda$ is a power of $p$.
Using the binomial theorem and the fact that $p$ divides $p^n\choose k$ for all $k\neq 0, p^n$ you see that
$$(x-y)^\lambda =x^\lambda -y^\lambda $$
which proves your point.
See this for more info.