Denote $s_1=x+y$ and $s_2=x^2+y^2$. We need to prove that over the finite field $ \mathbb{F}_2$ it is impossible to express $xy$ as a polynomial in $s_1$ and $s_2$.
My initial attempt is noticing that plugging in $x=y=1$ gives $1$ for $xy$ and $0$ for both $s_1$ and $s_2$, however I'm not sure how to proceed from there. Any help is appreciated.
Hint: If $P(x,y)$ can be written as a polynomial in $Q(x,y)$ and $R(x,y)$, then it is certainly a function of $Q(x,y)$ and $R(x,y)$ -- that is, you should be able to determine the value of $P(x,y)$ from the values of $Q(x,y)$ and $R(x,y)$. This means that, if I tell you $x+y$ and $x^2+y^2$, you should be able to tell me $xy$. Can you exhibit some choices of $(x,y)$ that agree on $x+y$ and $x^2+y^2$ but not at $xy$?