Let $K$ be a charateristic $p$ global field. For each place $P$ we let $K_P$ to be the completion at $P$ (which is a local field) and let $t_P$ be their prime. Let $t$ be a separating element of $K$ (which means $K/F(t)$ is a separable extension where $F$ is the constant field over $K$). […]

For each $n \geq 1$, we denote by $\mathbb{Q}_n$ the unique subfield of $\mathbb{Q}(\zeta_{p^{n+1}})$ for which $[\mathbb{Q}_n:\mathbb{Q}]=p^n$. Let now $K$ be a finite Galois extension of $\mathbb{Q}$ with the following properties: K contains $\mathbb{Q}_m$ for some $m \geq 1$; There exists an element $g \in Gal(K/\mathbb{Q})$ with the property that the order of $g$ is […]

Let $K$ be a finite extension of $\mathbf{Q}_p$. I have this vague intuition that $K$ having a lot of wild ramification is closely related to $K$ being close to containing high degree $p$-power roots of unity. I was wondering if this intuition is true and if maybe the following result (which would formalise it) is […]

Do you know how to compute a uniformizer of $\mathbb{Q}_p(\zeta_{p^n},p^\frac{1}{p})$? Where $\zeta_{p^n}$ is a primitive $p^n$-th root of 1 and $p$ is an odd prime.

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