On the relationship between Kurihara's classification and the theory of ramification removal (Q2849056)

Let \(K\) be a field of characteristic 0 which is complete with respect to a discrete valuation \(v_K\). Assume that the residue field \(\overline{K}=O_K/m_K\) of \(K\) is isomorphic to the field \({\mathbf F}_q((T))\) of formal Laurent series over a finite field \({\mathbf F}_q\) of characteristic \(p\). Then \(K\) is a local field of dimension 2. In [Compos. Math. 63, 237--257 (1987; Zbl 0674.12007)], \textit{M. Kurihara} classified such fields \(K\) into two groups based on properties of the completion \(\hat{\Omega}_{O_K/{\mathbb Z}}^1\) of the module of differentials \(\Omega_{O_K/{\mathbb Z}}^1\) with respect to the \(m_K\)-adic topology. Let \(\pi\) be a uniformizer for \(K\) and let \(t\) be an element of \(O_K\) whose image in \(O_K/m_K\) is a uniformizer for \(\overline{K}\). Then there are \(a,b\in O_K\), not both 0, such that \(a\,d\pi+b\,dt=0\). Kurihara defined \(K\) to be of type I if \(v_K(b)>v_K(a)\), and of type II if \(v_K(b)\leq v_K(a)\).NEWLINENEWLINEIn the paper under review the author shows that the quantity \(\Delta_K(\pi,t)=v_K(b)-v_K(a)\) is independent of the choice of \(a\) and \(b\). She also shows how the values taken by \(\Delta_K\) can be used to determine whether \(K\) is standard or almost standard. Say that \(K\) is ``standard'' if there is a finite extension \(k\) of the \(p\)-adic field \({\mathbb Q}_p\) such that \(K\) is isomorphic to NEWLINE\[NEWLINEk\{\{T\}\}=\left\{\sum_{n=-\infty}^{\infty} a_nT^n:a_n\in k,\;\inf v_k(a_n)>-\infty,\; \lim_{n\rightarrow-\infty}v_k(a_n)=\infty\right\}.NEWLINE\]NEWLINE Say that \(K\) is ``almost standard'' if there is a finite unramified extension \(K'/K\) such that \(K'\) is standard. The author shows that \(K\) is standard if and only if there are choices of \(\pi\) and \(t\) which make \(\Delta_K(\pi,t)=\infty\), and that \(K\) is almost standard if and only if there are choices of \(\pi\) and \(t\) which make \(\Delta_K(\pi,t)\) arbitrarily large. The proofs are based on finding a finite extension \(L/K\) such that \(L\) is a standard field and computing \(\Delta_K(\pi,t)\) in terms of derivatives with respect to a parameter \(t_L\) for \(L\).