What is algebraic element?

Let $L/K$ and $\alpha \in L$. If some nonzero $P \in K[X]$ has $P(\alpha) = 0$, $\alpha$ is algebraic over $K$; the monic generator $\pi_\alpha$ of the ideal $\{P : P(\alpha) = 0\}$ of $K[X]$ is its minimal polynomial, an irreducible polynomial ($\pi = QR$ with $Q(\alpha) = 0$ forces $R$ constant by minimality of the degree). Otherwise $\alpha$ is transcendental. We write $K(\alpha)$ for the smallest subfield of $L$ containing $K$ and $\alpha$, and $K[\alpha]$ for the smallest subring.