Is $t^3+t+1$ irreducible over $\mathbb{Q}(\sqrt{-31})$?

Question

To prove this, I substituted $a + b \sqrt{-31}$ into the polynomial, where $a,b$ are rational. But it is very complicated. Is there another way to check irreduciblity of $t^3+t+1$ over $\mathbb{Q}(\sqrt{-31})$?

Answer 1

A cubic polynomial $f$ that is irreducible over $\mathbb{Q}$ is irreducible over any quadratic extension of $\mathbb{Q}$.

To see this, let $\alpha$ be a root of $f$, and note that $\mathbb{Q}(\alpha)$ has degree $3$ over $\mathbb{Q}$, so it cannot be contained in an extension of degree $2$.