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 Post subject: UniquenessPosted: Sat, 10 Mar 2012 16:14:15 UTC
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Location: England
Suppose . Then, apparently, x can be uniquely written in the form
,
where the integers m and n are coprime to p, n>0 and is in lowest terms ( is then the order of x).

What I was wondering, though, was what about something like ?
Because I could write that:
1. , so the order is 2
or
2. , so the order is -2. But then I can't work out which part of the conditions this example is violating, because this isn't very unique.

Thank you!

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 Post subject: Re: UniquenessPosted: Sat, 10 Mar 2012 16:25:56 UTC
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Joined: Tue, 20 Oct 2009 18:18:45 UTC
Posts: 33
Location: Lisbon, Portugal
if you consider any rational number a/b:

a/b = (1/b)*a^1
a/b = a*(b^(-1))

so i think you must have an aditional condition so you can have a unique form to write them down... (for instance the order is also a non-negative number...)

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 Post subject: Re: UniquenessPosted: Sat, 10 Mar 2012 16:37:31 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
peccavi_2006 wrote:
Suppose . Then, apparently, x can be uniquely written in the form
,
where the integers m and n are coprime to p, n>0 and is in lowest terms ( is then the order of x).

What I was wondering, though, was what about something like ?
Because I could write that:
1. , so the order is 2
or
2. , so the order is -2. But then I can't work out which part of the conditions this example is violating, because this isn't very unique.

Thank you!

What you have is the p-adic valuation (or more generally, the order function at an irreducible prime of a factorial ring R whose field of fractions is F). The dependence on the prime p is implicit (and suppressed).

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 Post subject: Re: UniquenessPosted: Sat, 10 Mar 2012 16:38:10 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Moved from Algebra to A&NT.

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 Post subject: Re: UniquenessPosted: Sat, 10 Mar 2012 16:45:19 UTC
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Location: England
Oh!
So then , whereas ? I then assume that ?

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 Post subject: Re: UniquenessPosted: Sat, 10 Mar 2012 16:47:14 UTC
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peccavi_2006 wrote:
Oh!
So then , whereas ? I then assume that ?

Yes.

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 Post subject: Re: UniquenessPosted: Sat, 10 Mar 2012 16:49:07 UTC
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Location: England
Awesome - thanks outermeasure

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 Post subject: Re: UniquenessPosted: Sat, 10 Mar 2012 20:12:11 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12103
Location: Austin, TX
Voxx wrote:
if you consider any rational number a/b:

a/b = (1/b)*a^1
a/b = a*(b^(-1))

so i think you must have an aditional condition so you can have a unique form to write them down... (for instance the order is also a non-negative number...)

No, it depends on the choice of prime is all.

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