Re: JSH: Now a change

From: David Hartley (me9_at_privacy.net)
Date: 10/16/04 

Date: Sat, 16 Oct 2004 23:43:13 +0100

In message <3c65f87.0410161052.431e52f3@posting.google.com>, James
Harris <jstevh@msn.com> writes
>David Hartley <me9@privacy.net> wrote in message news:<NuqNToLyBIcBFw8N
>@212648.invalid>...

<snipped>

>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.

This is nonsense, if a factor exists it exists, not "appears to
meaningfully exist". If its inverse is not an algebraic integer then it
is not a unit in the algebraic integers; this isn't a "technicality",
it's straight from the definition.

>> 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?

If your ring is allowed to include transcendentals (as you have said
elsewhere) then it's easy. Let R(t) denote the smallest subring of the
complex numbers containing the algebraic integers and the complex number
t. R(t) fits your two requirements for any transcendental t, but so does
R(1/2t). Any larger ring containing both of these will contain the
element t*(1/2t) = 1/2 and so fails requirement 1. Hence there can't be
a unique largest ring fitting your requirements.

If you confine yourself to subrings of the algebraic numbers, the first
problem is to show that there are any such rings (apart from the ring of
algebraic integers). I can't see how to do that myself, but I found a
post by Arturo Magidin [1] where he claims it can be done using
valuations (no, I don't know what they are) and several posts referring
to a proof by K Ramsay, which I haven't tracked down. You seem to
believe this yourself, but I haven't seen you give any justification.
Once you've got an algebraic number a1, not an algebraic integer, such
that R(a1) fits your requirements then so, I suppose, do R(a2),...,R(an)
where a2,...,an are the other roots of the minimal polynomial of a1. But
any ring containing all these would have as elements all the elementary
symmetrical combinations of a1,...,an, at least one of which must be
rational and not an integer.

>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.

If you define a new mathematical object the onus is on you to prove it
exists.

[1] http://groups.google.com/groups?hl=en&lr=&frame=right&th=d4df579ba59
30db0&seekm=bmc569%241tvp%241%40agate.berkeley.edu#link14 

-- 
David Hartley

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