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Topic: FLT Discussion: Simplifying
Replies: 65   Last Post: Mar 17, 2001 11:59 PM

 Messages: [ Previous | Next ]
 hale@mailhost.tcs.tulane.edu Posts: 229 Registered: 12/8/04
Re: FLT Discussion: Simplifying
Posted: Jan 21, 2001 8:39 AM

In article <94cvf0\$4p6\$1@nnrp1.deja.com>,
jstevh@my-deja.com wrote:
> In article <3a683db5.481647214@news.newsguy.com>,
> randyp@visionplace.com (Randy Poe) wrote:

> > On Fri, 19 Jan 2001 11:24:12 GMT, "Dik T. Winter" <Dik.Winter@cwi.nl>
> > wrote:
> >

> > > > I'm looking for something that pushes me outside of integers
> because
> > > > that's what I want to prove must happen!
> > >
> > >Tsk. Your initial use of sqrt(-1) pushes you out of the integers

> >
> > You misunderstood this statement. He believes that the fact that
> > sqrt(-1) pushes him out of the integers is the contradiction and that
> > the proof can end at that point, just with the factorization. That
> > writing down a statement not in integers, though his first equation
> > was in the integers, is a contradiction.
> >

>
> Here's where there's the issue between what I've recently called
> patterns and regular rings.
>
> My understanding is that mathematicians have avoided this through
> coupling.
>
> That is, polynomials have counting number exponents *and* counting
> number coefficients.

Polynomials are not limited to just counting number coefficients.
I am sure that you are aware of this, so I don't know why you

> However, I have argued that this coupling is simply one way of doing
> things and there's no mathematical or logical requirement for it.

Polynomials in X can have coefficients in any ring R (which is then
fixed). The polynomials themselves then form a ring, called the
ring of polynomials in X and Y over R. This polynomial ring is
denoted by R[X, Y]. Of course, you can have more or less variables
than just X and Y and you need not even call them X, Y etc.

> The only remaining issue then has been an insistence that it must be
> proven that (x+sqrt(-1)y)(x-sqrt(-1)y) = 0 means that x+sqrt(-1)y = 0
> or x - sqrt(-1)y = 0, as some have insisted I must move to complex
> numbers to get this result.

No one has suggested that you *must* move to the complex numbers to
get this result. I suggested it as one way to get the ring that
you are working in to be a known one, so that you would not have
to prove that it exists and you could use many of the theorems already
proved about the complex numbers: for example, that it is a commutive
field with an isomorphic copy of the integers embedded in it.

> But, I've repeatedly brought up the fact that it's true for other
> rings, so why if you guys are acting like using rings is so fundamental
> and important, do you wish to make a result that's a hack depending on
> what ring you're using?
> Don't understand what I mean?

I don't understand what you mean. First, the *fact* is not true for
all rings. But, more importantly, as I have been trying to emphasize,
you need to state what ring you are working in so that 1) your
statements in your proof have meaning; 2) the words you use that have
been defined can be "unwind" to their more basic meaning; and 3) the
properties known to be true about that ring can be quoted (since
you seldom need to start from scratch with a ring that nobody has
ever thought of).

An example of 2 is the following. Suppose I say that 2 is prime.
If I want to "unwind" prime and eliminate that word, then I
cannot do it unless you tell me what ring you are working in.
For, 2 is prime in the integers, but 2 is a unit in the rationals,
and 2 = (1+i)(1-i) is composite in the gaussian integers. I need
to know what ring the statement "2 is prime" is referring to,
in order to know whether or not 2 has that property of primeness.

--
Bill Hale

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Date Subject Author
1/15/01 jstevh@my-deja.com
1/15/01 Dik T. Winter
1/16/01 Charles H. Giffen
1/16/01 jstevh@my-deja.com
1/16/01 Randy Poe
1/18/01 jstevh@my-deja.com
1/18/01 Michael Hochster
1/18/01 Peter Johnston
1/18/01 Randy Poe
1/18/01 Doug Norris
1/16/01 Doug Norris
1/16/01 Randy Poe
1/16/01 Dik T. Winter
1/18/01 jstevh@my-deja.com
1/19/01 Dik T. Winter
1/19/01 Randy Poe
1/20/01 jstevh@my-deja.com
1/20/01 oooF
1/21/01 hale@mailhost.tcs.tulane.edu
1/21/01 Peter Percival
1/21/01 Randy Poe
1/26/01 Franz Fritsche
1/19/01 gus gassmann
1/20/01 jstevh@my-deja.com
1/20/01 Doug Norris
1/26/01 Franz Fritsche
1/16/01 hale@mailhost.tcs.tulane.edu
1/16/01 Randy Poe
1/17/01 hale@mailhost.tcs.tulane.edu
1/18/01 jstevh@my-deja.com
1/19/01 hale@mailhost.tcs.tulane.edu
1/20/01 jstevh@my-deja.com
1/21/01 hale@mailhost.tcs.tulane.edu
1/18/01 Peter Percival
1/19/01 hale@mailhost.tcs.tulane.edu
3/17/01 Ross A. Finlayson
1/16/01 hale@mailhost.tcs.tulane.edu
1/18/01 jstevh@my-deja.com
1/19/01 hale@mailhost.tcs.tulane.edu
1/29/01 jstevh@my-deja.com
1/19/01 Dik T. Winter
1/21/01 Dennis Eriksson
1/15/01 Michael Hochster
1/16/01 jstevh@my-deja.com
1/16/01 Michael Hochster
1/18/01 jstevh@my-deja.com
1/18/01 Peter Percival
1/18/01 Randy Poe
1/19/01 oooF
1/21/01 Dik T. Winter
1/21/01 oooF
1/18/01 Edward Carter
1/19/01 W. Dale Hall
1/19/01 Michael Hochster
1/16/01 Randy Poe
1/16/01 Randy Poe
1/17/01 W. Dale Hall
1/17/01 W. Dale Hall
1/19/01 oooF
1/16/01 Charles H. Giffen
1/16/01 David Bernier
1/16/01 jstevh@my-deja.com
1/18/01 Arthur
1/30/01 plofap@my-deja.com
1/30/01 plofap@my-deja.com
1/30/01 plofap@my-deja.com