How do elements of $\mathbb{R}(xy,x+y)$ look like? I have problems with determining what are typical elements of such field $\mathbb{R}(xy,x+y)$
In one indeterminate it is easier as 
$\mathbb{R}(x)=\Bigl\{\frac{f(x)}{g(x)}, g(x)\neq 0, f(x),g(x)\in\mathbb{R}[x]\Bigr\} $
where 
$f(x)=a_{0}+a_{1}x+a_{2}x^2+\dots+a_{n}x^{n}$
$g(x)={b_{0}+b_{1}x+b_{2}x^2+\dots+b_{n}x^{n}}$
and by translating the above case, we get something like this
$\mathbb{R}(xy,x+y)=\Bigl\{\frac{f(xy,x+y)}{g(xy,x+y)}, g(xy,x+y)\neq 0, f(xy,x+y),g(xy,x+y)\in\mathbb{R}[xy,x+y]\Bigr\} $
but does $f(xy,x+y)$ really mean? Can we take for example
$f(xy,x+y)=5xy-3(x+y)$?
Can someone give me example of non trivial elements belonging to the above field?
For example is it true that  $(xy)^2=x^2y^2\in \mathbb{R}(xy,x+y) $?

HINT
You can "build these up". Notice that $\mathbb{R}(xy,x+y) = \mathbb{R}(xy)(x+y)$. 
This is similar to $\mathbb{R}(x+y)$ but instead of having coefficients in $\mathbb{R}$, the coefficients are in $\mathbb{R}(xy)$.
