Galois solvable and Galois abelian elements

Question

Let $F$ be a field of characteristic zero. Let $α$ be an element of some extension field of $F$ that is algebraic over $F$. Say that $α $ is Galois solvable over $F$ if $F(α)$ is Galois over $F$ and $Gal(F(α)/F)$ is a solvable group, and Say that $α$ is Galois abelian over $F$ if $F(α)$ is Galois over $F$ and $Gal(F(α)/F)$ is an abelian group.

Prove or give a counterexample:

1) If $α$ is Galois solvable over $F$ and $β ∈ F(α)$ then $β$ is Galois solvable over $F$.

2) If $α$ is Galois abelian over $F$ and $β ∈ F(α)$ then $β$ is Galois abelian over $F$.

Thanks.

Answer 1

• If $G:=\operatorname{Gal}(F(\alpha)/F)$ is solvable, there is even a priori no reason that the intermediate extension $F(\beta)/F$ is Galois. Take for example the splitting field of $X^3-2$ over $\mathbf{Q}$. It has solvable Galois group $S_3$, but $\mathbf{Q}(\sqrt[3]{2})/\mathbf{Q}$ is not Galois.

• If $G$ is abelian, then all intermediate extensions are Galois. By the Galois correspondence, $\operatorname{Gal}(F(\beta)/F)$ is a quotient of $G$, thus abelian too.