$K(ab,a+b) \subset K(a,b)\;$ finite field extension

Question

Let $\; K(ab,a+b) \subset K(a,b)\subset L \quad a,b \in L$

Is $\; K(ab,a+b) \subset K(a,b) \;$ a finite field extension and if not can anyone give a counterexample ?

Answer 1

$x^2 - (a+b)x + ab \in K(ab,a+b)[x]$ has roots $a$ and $b$ thus showing that $a,b \in K(a,b)$ are algebraic over $K(ab,a+b)$. Then adjoin both and get $K(ab,a+b,a,b) = K(a,b)$ is a finite extension.