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If you take the compositum $K=TP$ of the maximal tamely ramified extension $T$ of $\mathbf{Q}_p$ with a totally ramified extension the cyclotomic $P$ of \mathbf{Z}_p$-extension$\mathbf{Q}_p$P$ of degree $p$, \mathbf{Q}_p$, then$K$is not algebraically closed, its residue field is$\bar{\mathbf{F}}_p$, and the value group is$\mathbf{Q}$. 1 If you take the compositum$K=TP$of the maximal tamely ramified extension$T$of$\mathbf{Q}_p$with a totally ramified extension$P$of$\mathbf{Q}_p$of degree$p$, then$K$is not algebraically closed, its residue field is$\bar{\mathbf{F}}_p$, and the value group is$\mathbf{Q}\$.