An ultrametric space of Eisenstein polynomials and ramification theory (Q2846839)

Let \(K\) be a field which is complete with respect to a discrete valuation \(v_K\), with residue field \(k\) of characteristic \(p\). For \(e\geq1\) let \(E_K^e\) denote the set of monic Eisenstein polynomials over \(K\) with degree \(e\). For \(g(X)=\sum a_iX^i\) and \(h(X)=\sum b_iX^i\) in \(E_K^e\) define NEWLINE\[NEWLINEv_K(g,h)=\min\left\{v_K(a_i-b_i)+\frac{i}{e}: 0\leq i<e\right\},NEWLINE\]NEWLINE and let \(M_g\cong K[X]/(g(X))\) denote the extension of \(K\) generated by a single root of \(g(X)\). Let \(f\in E_K^e\) be such that \(L:=M_f\) is Galois over \(K\). Let \(m\in e^{-1}\cdot{\mathbb Z}\) and consider the following condition: NEWLINE\[NEWLINE (T_m^e)\;\;\text{If \(g\in E_K^e\) satisfies \(v_K(f,g)\geq m\) then \(M_g\) is \(K\)-isomorphic to \(L\).} NEWLINE\]NEWLINE In this paper the author gives necessary and sufficient conditions for \((T_m^e)\) to hold.NEWLINENEWLINELet \(r\) denote the largest upper ramification break of the extension \(L/K\). Let \(g\in E_K^e\) and set \(w=v_K(f,g)\). The author uses a result of \textit{J.-M. Fontaine} [Invent. Math. 81, 515--538 (1985; Zbl 0612.14043)] to show that if \(w<r+1\) then \(M_g\) is not \(K\)-isomorphic to \(L\), while if \(w>r+1\) then \(M_g\) is \(K\)-isomorphic to \(L\). Hence \((T_m^e)\) is true if \(m>r+1\), and false if \(p\mid e\) and \(m<r+1\). As for the case \(m=r+1\), let \(G^r\) denote the \(r\)th upper ramification group of \(G\) and let \(k^{sep}\) be a separable closure of \(k\). The author shows that \((T_{r+1}^e)\) is true if and only if there are no nontrivial continuous homomorphisms from Gal\((k^{sep}/k)\) to \(G^r\). In particular, if \(k\) is finite then \((T_{r+1}^e)\) is false, while if \(k\) is separably closed then \((T_{r+1}^e)\) is true.