**(a)** I want to use a result known as the "Subfield Test", but unfortunately I don't see how to make use of the fact that $\displaystyle char(F)=p$ in order to simplify things.

To use the Subfield Test I must show that for any $\displaystyle x,y$ $\displaystyle (y \neq 0)$ in K, $\displaystyle x-y$ and $\displaystyle xy^{-1}$ belong to $\displaystyle K$.

Let $\displaystyle x , y \in K$, so by definition $\displaystyle x^p =x$ and $\displaystyle y^p=y$. But how do we deduce that ($\displaystyle x-y)^p = (x-y)$ or $\displaystyle (xy^{-1})^p = xy^{-1}$?

The only thing we are told is the characteristic of F is a prime (which is the least positive integer p such that px=0 for all x in F). Is there any way we can somehow use Fermat's little theorem?

**(b)** Any clues to get me started is really appreciated.