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?
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?
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)
Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email> wrote: > In article <20091107144604.A50513F...@grex.org>, > p...@grex.org (Graven Water) wrote: >> 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.
When you complete the metric, is the resulting space also algebraically complete? And do you get a lot of nice properties that aren't true for analytic functions on the p-adics?
In article <20091109015420.3E66A3F...@grex.org>, p...@grex.org (Graven Water) wrote:
> Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email> wrote: > > In article <20091107144604.A50513F...@grex.org>, > > p...@grex.org (Graven Water) wrote: > >> 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.
> When you complete the metric, is the resulting space also algebraically > complete? And do you get a lot of nice properties that aren't true for > analytic functions on the p-adics?
I wish you wouldn't say "algebraically complete" when the standard term is "algebraically closed," but, yes, when you form the metric completion of the algebraic closure of the p-adic completion of the rationals, what you get is algebraically closed.
As for what properties you get, I refer you to Koblitz.
-- Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
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