If $K=F(K^p)$ is a finite extension and $\{a_1,\ldots,a_n\} \subset K$ linearly independent then so is $\{a_1^p,\ldots,a_n^p \}$ 
Suppose that $F$ is a field of characteristic $p$. Let $K/F$ be a finite extension and $K=F(K^p)$, where $K^p:= \{x^p\mid x\in K\}$. Suppose $\{a_1,\ldots,a_n\} \subset K$ is linearly independent over $F$. Show that is $\{a_1^p,\ldots,a_n^p \}$ is also linearly independent over $F$.

Here is a sketch of my proof: Suppose $b_1a_1^p+\cdots +b_na_n^p=0$ where $b_i\in F$. I want to find $c_i$ such that $c_i^p=b_i$. So $(c_1a_1+\cdots +c_na_n)^p=0$ and hence $c_i=b_i=0$. But the problem is that I don't know how to prove the existence of $c_i$. Any ideas?

First notice that every element of $F(K^p)$ is a linear combination of elements from $K^p$ with coefficients in $F$.
We may assume that $\{a_1,\dots,a_n\}$ is an $F$-basis of $K$. The remark above tells us that $\{a_1^p,\dots,a_n^p\}$ spans $F$ over $K$, so it is a basis.
