Let $K(\alpha)/K$ and $K(\beta)/K$ algebraic field extensions so that their respective Galois groups are abelian.
Prove that the Galois group of the field extension $K(\alpha + \beta)/K$ is also abelian.
I've tried considering the towers
$$K(\alpha,\beta)/K(\alpha)/K \qquad K(\alpha,\beta)/K(\beta)/K$$
Are somehow related to
$$K(\alpha,\beta)/K(\alpha+\beta)/K \qquad K(\alpha,\beta)/K(\alpha - \beta)/K$$
But I don't know how to relate this to the fact that the quotient is abelian or whether this statement is true or not.