= Krull dimension of rings of algebraic integers
{c}
{title2=$\dim\mathcal O_K=1$}
For any algebraic field extension $K/\mathbb Q$, including an infinite one, its ring $\mathcal O_K$ of elements integral over $\mathbb Z$ is an <integral extension> of $\mathbb Z$. The <Lying-over theorem>, <going-up theorem> and <incomparability theorem for integral extensions> give $\dim\mathcal O_K=\dim\mathbb Z=1$. The argument does not require that $\mathcal O_K$ be Noetherian or finitely generated over $\mathbb Z$.
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