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Let $K$ be a field and $R$ a valuation ring of $K$ with maximal ideal $m_R$. Let $a \in R$ such that $1-a \in m_R$.

Statement: For any $s$ that is not a multiple of the characteristic of $R/m_R$, the element $(1+a+a^2+\cdots+a^{s-1})^{-1}$ is inside $R$.

How do we prove the above statement?

Manos
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1 Answers1

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Reducing mod $m_R$, the element $1 + a + \cdots + a^{s-1}$ becomes $s$, which is invertible by assumption. Since $R$ is a local ring, any element of $R$ which is invertible mod $R$ is invertible in $R$, thus $1 + a + \cdots + a^{s-1}$ is invertible in $R$.

Matt E
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