Re: JSH: Now a change

From: James Harris (jstevh_at_msn.com)
Date: 10/16/04 

Date: 16 Oct 2004 11:52:15 -0700

David Hartley <me9@privacy.net> wrote in message news:<NuqNToLyBIcBFw8N@212648.invalid>...
> In message <3c65f87.0410151547.3158a4dc@posting.google.com>, James
> Harris <jstevh@msn.com> writes
> >David Hartley <me9@privacy.net> wrote in message
> >news:<9ovhCNz54wbBFwiY@212648.invalid>...
> >> In message <3c65f87.0410141349.77583757@posting.google.com>, James
> >> Harris <jstevh@msn.com> writes
> >>
> >> (much snipped)
> >>
> >> >Again, like I said at the top, the ring of objects is the largest ring
> >> >for which integers hold coprimeness.
> >>
> >> But coprime integers remain coprime in any larger ring, as has been
> >> pointed out to you several times before.
> >
> >Ok.
> >
> >If it will make you happy, then the object ring is the largest ring
> >where integers that are not factors of each other in the ring of
> >integers are still not factors in that ring.
> >
> >Notice that my two requirements cover that already:
> >
> >1. No rationals besides 1 and -1 are units in the object ring.
> >
> >2. No non-unit in the ring is a factor of any two integers that are
> >coprime in the ring of integers.
> >
> >If you were a mathematician, you would follow from my current
> >requirements, if you know them.
> >
> >If you missed them, then yeah, you have an alibi.
>
> Yes, I didn't take in your other definition.
>

Ok. Alibi accepted.

> >But if not, then you cannot be a mathematician, as your objection does
> >not apply given the full requirements, and as a mathematician, you
> >should have realized that fact.
> >
> >
> >> (If m and n are coprime integers, there exist integers a and b such that
> >> am + bn = 1. This equation is still true in the larger ring, so m and n
> >> are still coprime. If r is an element of the larger ring which divides m
> >> and n then it divides am + bn and so divides 1, i.e. it is a unit.
> >> However you expand the ring, you can't get non-unit common factors of m
> >> and n.)
> >>
> >> >Basically it's the largest ring that you can have and not conflict with
> >> >coprimeness results from the ring of integers.
> >> >
> >> >As an example of the opposite consider algebraic numbers, where you
> >> >break *every* coprimeness rule of the ring of integers as no integer is
> >> >coprime in the ring of algebraic numbers!
> >>
> >> All (non-zero) integers are coprime in this ring - it's a field.
> >
> >So you have a dumb definition of coprimeness.
>
> It seems to be the standard, quoted by several posters to these threads.
> It is equivalent, in the case of integers, to having no common, non-unit
> factor in the given ring. If you're talking about a different property
> from everyone else it would be helpful to call it something else.

Yes, but the problem is that algebraic integers allow a special
situation, where you have an appearance that a non-unit factor
meaningfully exists, though actuality it's simply not a unit in the
ring of algebraic integers because of a technicality i.e. its
muliplicative inverse is not an algebraic integer because it's not the
root of a monic polynomial with integer coefficients.

> Your requirement 2 is satisfied by any larger ring, requirement 1 only
> by any ring that contains no rationals other than the integers. So your
> object ring is the largest ring (presumably within the complex numbers)
> which contains the integers but no other rationals. For this to be
> well-defined you need to show that it is unique. I've just googled
> through some earlier threads and I see that this was pointed out to you
> at least a year ago, and it was demonstrated that there is no unique
> maximal ring fitting your requirements!
>

Well, what math arguments did they use?

You don't seem to understand how big the problem is.

If they used techniques that follow from the false assumption then
they didn't prove anything.

James Harris

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