Ramification in cyclic extensions of degree \(p^2\) of complete discrete valuation fields of prime characteristic \(p\) with imperfect residue field (Q2258052)

Let \(K\) be a complete discrete valuation field of characteristic \(p>0\) with discrete valuation \(v_K\) and residue field \(\bar{K}\). Let \(\pi\) be a prime element. A finite extension \(L/K\) is said \textit{unramified} if \(L/K\) is separable and \(v_K(\pi)=1\); \textit{wild} if \(L/k\) is a \(p\)-extension fully ramified and \textit{ferocious} if \(v_K(\pi)=1\) and \(L/K\) is purely inseparable. A cyclic extension \(L/K\) of degree \(p^n\) uniquely decomposes into a tower of cyclic extensions of degree \(p\): \(L= M_n/M_{n-1}/\cdots/M_1/M_0=K\). The \textit{genome} of \(L/K\) is the word \(T_1\cdots T_n\) where \[ T_i=\left\{\begin{matrix} \text{W}&\text{if}&M_i/M_{i-1}&\text{is wild}\cr \text{F}&\text{if}&M_i/M_{i-1}&\text{is ferocious}\cr \text{U}&\text{if}&M_i/M_{i-1}&\text{is wild}. \end{matrix}\right. \] The author considers a cyclic extension \(L/K\) of degree \(p^2\) with imperfect residue field \(\bar{K}\). The main result gives necessary and sufficient conditions for a pair of natural numbers \((h_1,h_2)\) to be the pair of lower ramification numbers of the extension \(L/K\). Namely, \((h_1,h_2)\) is such a pair iff one the following conditions holds: {\parindent=6mm \begin{itemize} \item[(1)] the genome is WW, \(p\nmid h_1, h_2\equiv h_1\bmod p, h_2 \geq (p^2-p+1)h_1\); \item [(2)] the genome is WF, \(p\nmid h_1, h_2> (p^2-p+1)\frac{h_1}{p}\); \item [(3)] the genome is FW, \(p\nmid h_2, h_2\geq (p^2-p+1)h_1\); \item [(4)] the genome is FF, \(h_2\geq (p^2-p+1)\frac{h_1}{p}\) and either \([\bar{K}:\bar{K}^p]>p\) or \([\bar{K}:\bar{K}^p]=p, p\mid h_1\). \end{itemize}}