# Ring of integers of p-adic field

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?

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 $$O_K=O_L[[\pi_K]]\cong O_L[[T]]/(f(T))$$.
• I never opened a book of $p$-adic number theory. – reuns 2 days ago
• Also, why do you have $\mathcal{O}_K = \mathbb{Z}_p[\zeta_{q-1},\pi_K]$? – marlasca23 2 days ago