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 Post subject: p-Adic valuationPosted: Fri, 24 Jun 2011 09:30:33 UTC
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Joined: Fri, 3 Sep 2010 09:29:45 UTC
Posts: 140
I've always wanted to learn p-Adic numbers, so I'm having a read through the notes at www.maths.gla.ac.uk/~ajb/dvi-ps/padicnotes.pdf.

I'm a little confused by the p-adic ordinal

1) They claim that . I presume this is obvious, but I just can't see it

2) I tried a test just to see how things work.

So this should be

Since 2 is prime

Since and we have that

Thus

But 1/5 does not divide 2/15. What am I doing wrong?

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 Post subject: Re: p-Adic valuationPosted: Fri, 24 Jun 2011 09:40:29 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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qwirk wrote:
I've always wanted to learn p-Adic numbers, so I'm having a read through the notes at www.maths.gla.ac.uk/~ajb/dvi-ps/padicnotes.pdf.

I'm a little confused by the p-adic ordinal

1) They claim that . I presume this is obvious, but I just can't see it

2) I tried a test just to see how things work.

So this should be

Since 2 is prime

Since and we have that

Thus

But 1/5 does not divide 2/15. What am I doing wrong?

You are thinking about it in the wrong way.

2/15 divided by 1/5 gives 2/3, which is a unit in . (Recall: is the localisation of at the prime ideal , i.e. )

We have the discrete valuation (where obviously for all nonzero a,b with ), and we extend this to a homomorphism of groups , so we also have for all .

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 Post subject: Posted: Fri, 24 Jun 2011 09:51:35 UTC
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Joined: Fri, 3 Sep 2010 09:29:45 UTC
Posts: 140
Ok so when we say we are not talking about divisibility over Z, but rather over some other integral domain?

(On a side note you just answered a lingering question I had - I often see the term "localised at a prime", but did note know what it meant)

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 Post subject: Posted: Fri, 24 Jun 2011 11:50:33 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6008
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
qwirk wrote:
Ok so when we say we are not talking about divisibility over Z, but rather over some other integral domain?

(On a side note you just answered a lingering question I had - I often see the term "localised at a prime", but did note know what it meant)

Sorry to confuse you further: depends on how you view things.

If you are thinking about , then it is divisibility over . However, this is not going to be very useful when you start doing more fun things because, for example, we want to construct the p-adic integers and do things over there or other overrings.

So, how can we do this? This is probably the place to introduce discrete valuation rings (DVRs) and valuation rings. A discrete valuation ring R is a PID with a unique maximal ideal, so we can put a discrete valuation with the properties:
and .

A valuation is when you relax v to take value in a totally ordered group, and a valuation ring is when you relax the PID to just being an integral domain R with valuation v on K=Frac(R) such that .

The maximal ideal allows us to define v by on , and we extend to by (check this is well-defined!). Of course, from this we can also recover --- note the we need the intersection with R here, otherwise you will end up with exactly where you are confused before --- because is not a valuation ring.

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 Post subject: Posted: Fri, 24 Jun 2011 14:43:36 UTC
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Joined: Fri, 3 Sep 2010 09:29:45 UTC
Posts: 140
Outermeasure - thanks for the detailed answer.

It might take a while to digest all that, but I appreciate it nonetheless!

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 Post subject: Posted: Sat, 25 Jun 2011 09:29:07 UTC
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Joined: Fri, 3 Sep 2010 09:29:45 UTC
Posts: 140
I think I also like the following definition (from "p-adic Numbers" by Fernando Q. Gouvea):

The p-adic valuation of is given by satisfying where

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