In a field $F=\{0,1,x\}$,  $x + x = 1$ and $x\cdot x = 1$ Looking for some pointers on how to approach this problem:

Let $F$ be a field consisting of exactly three elements $0$, $1$, $x$. Prove that 
  $x + x = 1$ and that  $x  x = 1$.


Write down the addition and multiplication tables. Much of them are known immediately from the properties of 0 and 1, and there's only one way to fill in the rest so that addition and multiplication by nonzero are both invertible.
