Explicit uniformizers for certain totally ramified extensions of the field of p-adic numbers

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Let p be an odd prime number. We construct explicit uniformizers for the totally ramified extension $${\mathbb {Q}}_p(\zeta _{p^2},\root p \of {p})$$ of the field of p-adic numbers $${\mathbb {Q}}_p$$ , where $$\zeta _{p^2}$$ is a primitive $$p^2$$ -th root of unity.