Let $k$ be a field of characteristic zero, $R$ a commutative $k$-algebra which is an integral domain having field of fractions $Q(R)$ and $a$ an algebraic element over $R$ ($a$ belongs to some algebraic closure of $Q(R)$, see this question).
Let $h=h(T) \in k[T]$ be a separable polynomial of degree $d \geq 2$, namely, $h$ has $d$ distinct roots in $\bar{k}$ (= an algebraic closure of $k$).
Recall the definition of a separable ring extension; according to wikipedia: "Any separable extension A / K of commutative rings is formally unramified. The converse holds if A is a finitely generated K-algebra".
Is it true that $A:=R[h(a)] \subseteq R[a]:=B$ is always a separable ring extension? If not, is it possible to characterize all $h$'s for which such extension is separable? What about flatness?
Notice that $B$ is a finitely generated $A$-algebra.
Special case: Is $\mathbb{Q}[\sqrt{3}] \subset \mathbb{Q}[3^{\frac{1}{4}}]$ separable? here $k=R=\mathbb{Q}$, $a=3^{\frac{1}{4}}$, $h=h(T)=T^2$.
Remark: Maybe S.S.S Wang's paper (Corollary 8,10) about separability is relevant here.
Any hints and comments are welcome! Thank you.