Let $\mathbb{Q}_p$ denote the field of p-adic numbers. For a prime number $q$ ($\neq p$), does exist a totally ramified extension $K/\mathbb{Q}_p$ with Galois group isomorphis to $\mathbb{Z}_q \times \mathbb{Z}_q$?
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1$\begingroup$ Yes if $q\nmid p-1$, otherwise no: $p\nmid [K:\Bbb{Q}]$ means that $K$ is a tamely ramified extension, thus $K\subset T=\bigcup_{p\nmid n}\Bbb{Q}_p(\zeta_n,p^{1/n})$, and $T\cap \Bbb{Q}_p^{ab}=\bigcup_{p\nmid n}\Bbb{Q}_p(\zeta_n,p^{1/(p-1)})$ (for $p$ odd) $\endgroup$– reunsCommented Oct 15, 2020 at 10:43
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2$\begingroup$ I missed you said totally ramified, thus always no. $\endgroup$– reunsCommented Oct 15, 2020 at 10:50
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5$\begingroup$ No, because the tame inertia group is cyclic. $\endgroup$– Chris WuthrichCommented Oct 15, 2020 at 11:09
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