# Tagged Questions

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### Definition of $\mathbb Q^c_p$

Let $p$ be a prime number and let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$, i.e. the field of algebraic numbers. Is it possible at all to define the $p$-adic completion ...
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### Lower Bound on Number of Wildly Ramified Extensions of $\mathbb{Q}_p$

Suppose $p$ is prime and $p$ divides $e$ divides $n$. Typically up to isomorphism there are a lot of wildly ramified extensions of $\mathbb{Q}_p$ which have degree $n$ and ramification index $e$. ...
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### Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
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### Image of the Norm on a Finite Dimensional Extension of $\mathbb{Q}_p$

I've been trying to see whether following assertion is true in order to give a quick proof of another problem I was doing: if $K$ is a finite dimensional extension of the $p$-adic numbers ...
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### Is there an explicit embedding from the various fields of p-adic numbers $\mathbb{Q}_p$ into $\mathbb{C}$?

For any field of p-adic numbers $\mathbb{Q}_p$, one can construct the field $\mathbb{C}_p$, the metric completion of one of its algebraic completions. By the axiom of choice, we can prove this to be ...
### Powers of $\Bbb Z_p\cap \Bbb Q$ on $1+x\Bbb F_q[[x]]$
Set $q=p^r$ for the finite field $\Bbb F_q$. In the formal power series ring $\Bbb F_q[[x]]$, there is a notion of convergence given by the underlying $(x)$-adic topology. If ...