Lemma 29.46.4. Let $A \subset B$ be a ring extension. If there exists $b \in B$, $b \not\in A$ and an integer $n \geq 2$ with $b^ n \in A$ and $b^{n + 1} \in A$, then there exists a $b' \in B$, $b' \not\in A$ with $(b')^2 \in A$ and $(b')^3 \in A$.

Proof. Let $b$ and $n$ be as in the lemma. Then all sufficiently large powers of $b$ are in $A$. Namely, $(b^ n)^ k(b^{n + 1})^ i = b^{(k + i)n + i}$ which implies any power $b^ m$ with $m \geq n^2$ is in $A$. Hence if $i \geq 1$ is the largest integer such that $b^ i \not\in A$, then $(b^ i)^2 \in A$ and $(b^ i)^3 \in A$. $\square$

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