Multiplicative inverse of (x-y) in a Field

Question

Consider $x, y \in F$ where F is a Field and $0 \neq x \neq y \neq 0$. What is the multiplicative(!) inverse of $(x-y) \in F$ ?

I'm struggling to work this out...

Answer 1

If $y = -x$ (which is possible under the assumption that $x \neq y$ only if $char(F) \neq 2$), then $(x - y)^{-1} = (2x)^{-1}$. Else $y \neq x, -x$ and so $y^2 \neq x^2$. Then $(x - y)^{-1} = (x + y)/(x^2 - y^2)$.

Answer 2

Unless $F$ is specified, the only reasonable answer is: it is the unique element $u\in F$ satisfying $(x-y)u=1$.

You cannot expect any general formula working for any field.