Is $\mathbb{Q}_2$ complete?

Question

Is the $p$-adic field $\mathbb{Q}_p$ complete respect to the $p$-adic norm $|\cdot|_p$ when $p=2$?

Answer 1

Why you went to ask this question depends on which definition of $\Bbb{Q}_p$ you started with, but the answer doesn't : there are 3 definitions

• one from $\Bbb{Z}_p=\varprojlim \Bbb{Z/p^n Z}$ (limits of sequences of integers that converge $\bmod p^n$ for all $n$, ie. for $|.|_p$) and $\Bbb{Q}_p=\Bbb{Z}_p[p^{-1}]$

• one from the rules of addition/multiplication on $p$-adic series $\sum_{n\ge -N} a_n p^n,a_n\in 0\ldots p-1$

• one as the completion of $\Bbb{Q}$ for the $p$-adic absolute value (similar to that $\Bbb{R}$ is the completion of $\Bbb{Q},|.|_\infty$)

Thus the answer is to prove that they are all equivalent. The reasoning doesn't depend on the chosen $p$ prime.