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My tutor recently mentioned the following result:

"For any complete discretely valued extension $K$ of $\mathbb{Q}_p$ with perfect residue field $k$, the ring of integers $\mathcal{O}_K$ can be written as a quotient of $W(k)[[T]]$ by a principal ideal generated by an Eisenstein polynomial"

I have tried looking at various books on local fields but this has not helped. Many hours of Internet search have not helped either. Could someone please tell me where I could find a proof of this fact?

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When $k$ is a finite field (ie. $K/\Bbb{Q}_p$ is algebraic) then $W(k)$ is a fancy way to say $\Bbb{Z}_p[\zeta_{q-1}]$ where $q=|k|$, the ring of integers of the largest unramified extension $\subset K$.

$\zeta_{q-1}\in K$ by Hensel lemma and $$O_K=\Bbb{Z}_p[\zeta_{q-1}][[\pi_K]]=\Bbb{Z}_p[\zeta_{q-1},\pi_K]\cong \Bbb{Z}_p[\zeta_{q-1}][T]/(f(T))$$ where $\pi_K$ is a generator of the maximal ideal of $O_K$ and $f(T)\in \Bbb{Z}_p[\zeta_{q-1}][T]$ is its minimal polynomial, which is Eisenstein, because $\pi_K^e = pu$ for some $e$ and $u\in O_K^\times$,

from which $\deg(f)= [K:\Bbb{Q}_p[\zeta_{q-1}]]=e$, which follows from $$O_K=\{ \sum_{j\ge 0} c_j \pi_K^j, c_j\in 0\cup \{\zeta_{q-1}^l\}\} = \sum_{j=0}^{e-1}\pi_K^j \Bbb{Z}_p[\zeta_{q-1}]$$

When $k$ is a more general residue field then the idea is the same: by Hensel lemma the unramified polynomials have roots in $K$ so there is a largest unramified extension $L$ such that $O_L/(p)=k$ and $O_K=O_L[[\pi_K]]\cong O_L[[T]]/(f(T))$.

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  • $\begingroup$ Thank you! I want to include a reference for this. Do you know any textbook where I could find this? $\endgroup$ – marlasca23 2 days ago
  • $\begingroup$ I never opened a book of $p$-adic number theory. $\endgroup$ – reuns 2 days ago
  • $\begingroup$ Also, why do you have $\mathcal{O}_K = \mathbb{Z}_p[\zeta_{q-1},\pi_K]$? $\endgroup$ – marlasca23 2 days ago

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