I'm reading through "$p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions" by Koblitz to learn about p-adic numbers. In chapter 3, he describes the construction of $\Omega$ (a.k.a. $\Omega_p$), the completion of the algebraic closure $\overline{\mathbb Q}_p$ of $\mathbb Q_p$. I think this is also called $\mathbb C_p$.

Could one obtain (a field isomorphic to) $\Omega$ more directly by simply completing $\overline{\mathbb Q}$ with respect to the $p$-adic norm $|~~|_p$? The way described in Koblitz starts with $\mathbb Q$ and the norm $|~~|_p$, completes it, constructs the algebraic closure, and then completes it again.

More specifically, the norm $|~~|_p$ should extend from $\mathbb Q$ to $\overline{\mathbb Q}$ in the same way Koblitz describes it extending from $\mathbb Q_p$ to $\overline{\mathbb Q}_p$ (using $\mathbb N_{\mathbb Q(\alpha) / \mathbb Q}$). Then completing $\overline{\mathbb Q}$ with respect to $|~~|_p$ should preserve the property of being algebraically closed using essentially the same proof.

Will this work, or will something go wrong?

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    $\begingroup$ The problem with the approach you suggest is that you forgot to define $|\cdot|_p$ on $\overline{\mathbf Q}$. It's true that $\overline{\mathbf Q}$ is dense in ${\mathbf C}_p$ (same thing as $\Omega_p$), so in a sense ${\mathbf C}_p$ is a $p$-adic completion of $\overline{\mathbf Q}$, but good luck giving a direct definition of $|\cdot|_p$ in one stroke on $\overline{\mathbf Q}$ without using ${\mathbf C}_p$. In case you don't see what I'm getting at, please tell us what $|1+2i|_5$ and $|1-2i|_5$ are. $\endgroup$ – KCd Jan 11 '14 at 3:02
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    $\begingroup$ Your idea that $|\cdot|_p$ extends from the non-complete field ${\mathbf Q}$ to its algebraic extension by using the norm is incorrect. Look at the proof that the definition of the extension of $|\cdot|_p$ from ${\mathbf Q}_p$ to its finite extensions using the norm is an absolute value and figure out where it breaks down if you replace ${\mathbf Q}_p$ with $\mathbf Q$. $\endgroup$ – KCd Jan 11 '14 at 3:07
  • $\begingroup$ Thanks for your speedy answer. The definition I had in mind was $|\alpha|_p = |\mathbb N_{\mathbb Q(\alpha) / \mathbb Q}(\alpha)|_p^{1/[\mathbb Q(\alpha) : \mathbb Q]}$ for $\alpha \in \overline{\mathbb Q}$. I see why the arguments in Koblitz' book do not apply here, but I'm curious if there is an example showing that what I'm proposing is not a norm on $\overline{\mathbb Q}$. $\endgroup$ – Zachary Jan 13 '14 at 18:22
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    $\begingroup$ Exactly. The point is that $|\cdot|_p$ has a unique extension from ${\mathbf Q}_p$ to any finite (or algebraic) extension, but this is false for finite extensions of ${\mathbf Q}$. In fact, the number of extensions of $|\cdot|_p$ from ${\mathbf Q}$ to a finite extension $K/{\mathbf Q}$ is equal to the number of prime ideals in ${\mathcal O}_K$ that lie over $p$. In particular, when $K = {\mathbf Q}(i)$ there are two extensions of $|\cdot|_p$ to $K$ when $p \equiv 1 \bmod 4$. $\endgroup$ – KCd Jan 13 '14 at 19:40
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    $\begingroup$ The two extensions of $|\cdot|_5$ to ${\mathbf Q}(i)$ are related to the two primes $1+2i$ and $1-2i$ that divide $5$. For one of the extensions, $|1+2i| = 1/5$ (not $1/\sqrt{5}$) and $|1-2i| = 1$, while for the other extension $|1-2i| = 1/5$ and $|1+2i| = 1$. These are the $(1+2i)$-adic and $(1-2i)$-adic absolute values. See math.uconn.edu/~kconrad/blurbs/gradnumthy/ostrowskiQ(i).pdf for a description of all the nontrivial absolute values on ${\mathbf Q}(i)$, up to equivalence. $\endgroup$ – KCd Jan 13 '14 at 19:43

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