I'm trying to learn Galois theory on Coursera. The lecturer gave the following definition of a stem field:
Let $P$ be an irreducible monic polynomial in $K[X]$ with a root $\alpha$. A stem field is an extension E such that $\alpha \in E$ and $E=K(\alpha)$.
A textbook I have says $\mathbb{Q}(\sqrt[3]{2})$ is not a stem field for the polynomial $X^3-2 \in \mathbb{Q}[X]$, but to me it seems to match the definition. Can someone help me understand why it isn't?