Can $x^2+xy+y^2$ be written using only $x-y$?

Question

My question is very specific and as such: Can $x^2+xy+y^2$ be written using only $x-y$, that is, as a function of $x-y$? Thanks.

Answer 1

No, it can't.

Suppose such a function F existed. Then for all x,y we would have $x^2+xy+y^2=F(x-y)$.

But then when $x=y$, we would have $3x^2=F(0)$, which is impossible. The LHS isn't constant while the RHS is.

EDIT: as pointed out in the comments, the previous argument doesn't work if the field is of characteristic $3$. In fact, the statement is true: in a field of characteristic $3$, we have $1=-2$, so $x^2+xy+y^2=x^2-2xy+y^2=(x-y)^2$.