Proposition 29.46.8. Let $A \subset B$ be a ring extension. The following are equivalent

$\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is a universal homeomorphism, and

every finite subset $E \subset B$ is contained in an extension

\[ A[b_1, \ldots , b_ n] \subset B \]such that for $i = 1, \ldots , n$ we have

$b_ i^2, b_ i^3 \in A[b_1, \ldots , b_{i - 1}]$, or

there exists a prime number $p$ with $pb_ i, b_ i^ p \in A[b_1, \ldots , b_{i - 1}]$.

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