Given $K(\alpha)/K$ and $K(\beta)/K$ abelian extensions, prove that $K(\alpha + \beta)/K$ is an abelian extension.

Question

Problem:

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.

My attempt:

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.

Answer 1

$K(\alpha, \beta)$ is Galois over $K$ and the corresponding Galois group is a subgroup of $\text{Gal}(K(\alpha)/K)\times \text{Gal}(K(\beta)/K)$. So $K(\alpha, \beta)/K$ is abelian. Thus, any intermediate Galois extension must be abelian since the corresponding Galois group would be a quotient of $\text{Gal}(K(\alpha,\beta)/K)$