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Let $k$ be a field and $|\cdot|$ is a absolute value of $k$. Let $\bar k$ be a completion of $k$. We define the ring of integers

$$ O=\{a \in k : |a|\leq 1\} $$

and define an ideal

$$ P=\{a \in k : |a|<1\}. $$

$P$ is a maximal ideal of $O$, so $O/P$ is a field. Now define $\bar O$ and $\bar P$ in a similar way in $\bar k$.

My question is : why $\bar O/\bar P$ is isomorphic to $O/P$? I have no ideal how to prove this.

(I'm studying Cassels and Frohlich's book, and they say that it is clear.)

LWW
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We have the map $O \to \bar{O}/ \bar {P}$, the kernel is clearly $\bar{P}$, so we wish to show it is onto.

The point is $O$ is dense in $\bar{O}$, and so the image is dense in $O \to \bar{O}/ \bar {P}$, but the latter has the discrete topology.

You can see this more explicitly if you want: given $o \in \bar O$, find $o'\in O$ that is very close to it, then $|o-o'| < 1$, so $o$ maps to $[o']$ like we wanted.

Andy
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    Thank you Andy! Now I understand – LWW Oct 14 '19 at 13:20
  • @LWW Np, enjoy learning about local fields! I do not know you, but to me it was motivating (I learned about them recently as well) to see cool applications that are easily attainable with a bit of study. Once you'll know about the Newton polygon, show that truncating the taylor expansion of $e^x$ gives an irreducible polynomial. See also https://math.stackexchange.com/questions/696669/can-one-show-a-beginning-student-how-to-use-the-p-adics-to-solve-a-problem – Andy Oct 14 '19 at 13:24