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Gerry Myerson  
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 More options Nov 9 2009, 12:57 am
Newsgroups: sci.math
From: Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email>
Date: Mon, 09 Nov 2009 11:57:34 +1100
Local: Mon, Nov 9 2009 12:57 am
Subject: Re: algebra and analysis
In article <20091107144604.A50513F...@grex.org>,
 p...@grex.org (Graven Water) wrote:

> The complex numbers are algebraically complete.  

> And complex analysis has nice theorems that real analysis doesn't have:  
> like, if a function is differentiable once, it's differentiable infinitely
> many times.  

> Are those two things related?

No. The algebraic closure of the rationals isn't a nice place
to do analysis.

> Is something similar true for p-adic analysis - does analysis over the
> algebraic completion of the p-adics have nice theorems?  if you could
> define a metric on the algebraic completion?

You can define such a metric, and then you can complete the metric,
and on that space, yes, people do analysis. Koblitz has written
a couple of helpful books on the topic.

--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)


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