What is the degree and minimal polynomial over $\mathbb{Q}$ of $j\sqrt{2}$, where $j$ denotes $-\frac12+\frac{\sqrt3}2i$, a primitive thrid root of unity?
I see that $(j\sqrt{2})^6=8$. Then $P(j\sqrt{2})=0$ with $P=X^6-8$. Can we say that $P$ is irreducible polynomial over $\mathbb{Q}$?