is the localisation of the ring $$A:=\mathbb{Z}_p[T]/(pT^2+T+1)$$ at the prime ideal (p) isomorphic to $\mathbb{Z}_p$?
If not, how to understand this ring very explicitly?
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$\begingroup$ EGA II 6.2.5, 6.2.6 give a decomposition to quai-finite schemes, I am looking for a concrete example in order to understand such decomposition, the above is my example. According to EGA, $A_p$ is finite over $\mathbb{Z}_p$, and it seems very likely $A_p=\mathbb{Z}_p$ $\endgroup$– HeerCommented May 16, 2012 at 20:54
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1 Answer
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That equation has a solution in $\mathbb Z_p$. Proof: $f(a)=-pa^2-1$ is a contraction mapping for the $p$-adic metric, and a fixed point must be a solution.
Let $\alpha$ be that solution, then $(T-\alpha)$ divides your equation. The other factor must be $pT-\alpha^{-1}$. So we have:
$A = \mathbb Z_p[T]/(T-\alpha) \times \mathbb Z_p [T]/(pT-\alpha^{-1})=\mathbb Z_p \times \mathbb Q_p$
The localization at $p$ is $\mathbb Z_p$.
answered May 16, 2012 at 21:24
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6$\begingroup$ Another way to see immediately that the equation has a solution is via Newton's polygon: it is formed by two sides with slopes 0 and 1, and each corresponds to a non-trivial factor, having degree 1. $\endgroup$ Commented May 16, 2012 at 21:45
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4$\begingroup$ Although I always prefer Newton poly., factorization follows also from Hensel’s Lemma, with the one polynomial in char. $p$ being $1+x$ and the other being $1$. HL guarantees a factor of degree $1$. $\endgroup$– LubinCommented May 17, 2012 at 2:27