A problem from Isaacs's book

Question

Suppose that F ⊆ E ⊆ L and suppose α ∈ L is algebraic over F . Let f := min E (α) (the minimal polynomial of α over E) I need to show all roots of f in any field extension of L are algebraic over F .

Can anyone help with this one, thanks

Answer 1

Hints:

Let $\;g(x)\in\Bbb F[x]\;$ be the minimal polynomial of $\;\alpha\;$ over $\;\Bbb F\;$. Since clearly $\;f(x)\,,\,g(x)\in \Bbb E[x]\;$ , we have that

$$f\mid g\;\;\text{in}\;\;\Bbb E[x]\iff g(x)=f(x)h(x)\;,\;\;h(x)\in\Bbb E[x]$$

and the last equality means any root of $\;f\;$ is also a root of $\;g\;$...