[Math] Minimal polynomial over $\mathbb Q(\sqrt{-2})$

Question

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}$

Answer 1

Suppose $f$ is an irreducible polynomial in $\Bbb Q[X]$, and it has a nontrivial irreducible factor $g$ over $K = \Bbb Q(\sqrt{-2})$.

$g$ can't have rational coefficients, so its conjugate $\overline g$ is another factor of $f$ over $K$. Since $g$ and $\overline g$ are coprime, $g \overline g$ is also a factor of $f$, but it has rational coefficients, so you must have $f = \lambda g \overline g$, and the degree of $f$ must be even.

In your case, $f$ has degree $3$ so this is impossible : it has to stay irreducible over $K$.