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) $?
This is the field of all rational functions symmetric in $x$ and $y$. So for example it contains $x^3+y^3+5x^2y+5xy^2 -x-y$ since this is symmetric in $x$ and $y$.
$x+y$ and $xy$ are the elementary symmetric functions and a basic theorem of algebra is the every symmetric rational functions is a rational function of the elementary symmetric functions.