$ab$ and $a+b$ algebraic implies $a,b$ algebraic

Question

$L/F$ be a field extension. Is it possible that $F (a,b)/F (ab,a+b)$ to be algebraic extension? I've tried to find a field $K$ $F (a,b)/K$ is algebraic extension and $K$ is subfield of $F(ab,a+b)$. Fortunately if $a, b$ are algebraic over $F$, $F$ could be such $K$. But I cannot find such $K$ for any $a,b$. Is this kind approach wrong?

Answer 1

Note that $a,b$ are roots of the quadratic polynomial $X^2-(a+b)X+ab$. Hence $F(a,b)/F(ab,a+b)$ is algebraic of degree $1$ or $2$.