Minimal polynomial over $\mathbb Q(\sqrt{-2})$ 
Find the minimal polynomial for $\sqrt[3]{25} - \sqrt[3]{5} $ over  $\mathbb Q$ and $\mathbb Q(\sqrt{-2})$.

I have done the first part of this, over $ Q$, and have a polynomial. But I do not know how to do this over $ Q \sqrt{-2}$

Put $\theta=\sqrt[3] 5$  so $$x=\theta(\theta-1)\Rightarrow x^3=\theta^3(\theta^3-3\theta^2+3\theta-1)\Rightarrow x^3+15x-20=0$$ This is the minimal polynomial (irreducible by Einsenstein with $p=5$) of $\theta(\theta-1)$ over $\mathbb Q$ and also the m. p. of the same element $\theta(\theta-1)$ over any $K=\mathbb Q(t)\supset \mathbb Q$ providing that it is  irreducible in $K[x]$.
We can use multiplication of degrees but (another way), splitting in $\mathbb C$ one has $$x^3+15x-20=(x-x_1)(x-\alpha)(x-\bar\alpha)$$ where
 $$x_1=\theta(\theta-1)\space\text {and}\space \alpha=\frac 12\left((1-i\sqrt3)\theta-(1+i\sqrt3)\theta^2\right)$$ hence it is directly seen that $x^3+15x-20$  is not decomposable on $\mathbb Q(\sqrt{-2})[x]$
