Let $E$ be a finite normal extension of the field $F$, for which the Galois group $G(E/F)$ is Abelian, and let $K$ be a field intermediate to $F$ and $E$. Let $a\in K$ and $f(x)\in F[x]$ be the minimal polynomial for $a$ over $F$, if $b\in E$ with $f(b)=0$. show that there is a polynomial $g(x)\in F[x]$ such that $g(a)=b$
I do not how to use the abelian condiction, i try to consider the question from the following two cases
(1) the character of $F$ is 0
(2) the character of $F$ is a prime $p$, but i still do not know how to prove this question.
You want to prove that $F\subset K$ is a normal extension. Since all extensions are Galois, this amounts to requiring that every subgroup of $G(E/F)$ is normal - which is true since $G(E/F)$ is abelian!