Showing that $f(x) = \frac{1}{2} x^3 + 3x^2 -\frac{5}{3} x - 5$ irreducible over $\mathbb{Q}(\sqrt[4]{5})$ I am unsure how to show that the polynomial $f(x) = \frac{1}{2} x^3 + 3x^2 -\frac{5}{3} x - 5$ is irreducible over $\mathbb{Q}(\sqrt[4]{5})$. If it were reducible, it would have a root in $\mathbb{Q}(\sqrt[4]{5})$. I need to pick a general element of the field, use the polynomial to get some nice conditions on it. That will give me a finite number of candidates for roots. Then I can check that none of the candidates is a root. I would appreciate any help to do this. Thanks.

Hint The Rational Root Test shows that $f$ is irreducible over $\Bbb Q$ so its Galois group has an element of order $3$. On the other hand, the Galois group of the minimal polynomial $x^4 - 5$ of $\sqrt[4]{5}$ over $\Bbb Q$ is $D_8$, which has order $8$.
