Re: JSH: Simple proof

From: Hagen (knaf_at_itwm.fhg.de)
Date: 12/02/04 

Date: Thu, 2 Dec 2004 14:03:29 +0000 (UTC)

On 01 Dec 2004, James Harris wrote:

>knaf@itwm.fhg.de (Hagen) wrote in message news:<200412010856.iB18uUI05300@proapp.mathforum.org>...

>> Dear James Harris,
>>
>> Why don't you simply and briefly state the >>problematic<< property
>> of the ring of algebraic integers and/or the statement that you
>> want to prove.

>
>The ring of algebraic integers is determined by roots of *monic*
>polynomials with integer coefficients.
>
>It is possible to show with basic algebra that there are numbers which
>are properly units but because their multiplicative inverse is not the
>root of some monic polynomial with integer coefficients they are not
>units in the ring of algebraic integers.
>

These numbers are >>properly units<< in which ring? The notion
of a unit is defined only with respect to some ring. Your
statement makes no sense as it stands.

>To see how it works consider that in rationals you can have
>
>(3x + 1)(x + 1) = 3x^2 + 4x + 1
>
>where, of course, one of the roots is a unit in the ring of algebraic
>integers.
>
>But now consider
>
>(3x + u_1)(x + u_2) = 3x^2 + kx + 1, where u_1 u_2 =1, and k is an
>integer.
>
>You find that if the u's are irrational, then u_1, while an algebraic
>integer is not a unit in the ring, while u_2 cannot then even be an
>algebraic integer.
>
>My research shows though that both u_1 and u_2 can be units in a ring
>where -1 and 1 are the only rational units, and no non-unit member of
>the ring is a factor of any two integers that are coprime in the ring
>of integers.
>
>You see, I abstracted out two key properties of rings like the ring of
>integers and the ring of algebraic integers.
>

I read your contributions about the >>object ring<< as well as the
ones of many others. Together we reached a rather satisfying
answer concerning the existence of such a ring.

>
>> Using standard mathematical terms and notions (for example from
>> commutative algebra) this should be possible in a few lines instead
>> of making a long story.
>
>It's not complicated. Basically you can't just rely on whether or not
>some number is in the ring of algebraic integers when considering
>factors of roots of a polynomial.
>
>The mathematics is mostly REALLY simple.
>

This is no clear mathematical statement at all!
What is the problem? What did you prove?

>> Why do we have to discuss things like >>what is a polynomial?<<
>
>I'm not discussing that, other posters made a big deal out of it.
>

...but if such simple things like >>what is the constant term
of a polynomial<< are not clear, we have to clarify it before
we can proceed.

>> here? The notion of a polynomial is defined since a long time and
>> can be found in every introductory book on algebra.
>
>So?
>
>
>> If we consistently use the common definitions of mathematical
>> objects like polynomials we should rather quickly be able to
>> clarify the situation and avoid all the frustration that frequently
>> seems to culminate in personal attacks.
>>
>> H
>
>I've seen posters come and go, and every once in a while there's a
>poster like you who claims to care about working things out.
>
>When it turns out that I'm right, you go over to the other side, and
>either run away, or turn to bizarre behavior.
>

I follow the discussions you initiated since almost one year
now and tried to contribute with the aim to clarify things
(indeed!). I wrote several contributions concerning the object
ring, because I found the idea interesting. However I remember
clearly that you left the discussion suddenly claiming that
you are not interested in the stuff anymore. Strangely you
did this when with the help of other posters together we
really reached a good overview over the candidates for >>object
rings<< within the field of algebraic numbers (and even within
larger fields).
No >>sides<< in that discussion just mathematics!

>
>Psychologists call it cognitive dissonance.
>
>Basically, deep down you believe that I must be wrong, so your post is
>not really in good faith. But simply *saying* certain things that
>indicate objectivity or willingness to be objective sets you up
>psychologically.
>
>That is, you feel a need to be consistent with what you said.
>
>But later, when you run into the rigid mathematics, which goes against
>what you wish to believe, you basically kind of break. Your mind
>breaks, and you run away or behave weird.
>
>I've seen it lots of times. Do yourself a favor, and just walk away
>now.
>
>
>James Harris

What sometimes really makes me angry in the discussions you
initiate is the lack of mathematical rigor! Each time one of
the participants clearly worked out a point the discussion
stops, or turns into personal attacks etc. but it never ends
with a statement like: now we have seen that this and this
is true. Lets keep this result and proceed until we reach
the final conclusion.

Why don't you answer to Nora Baron's recent posts? They are
mathematically very clear. The critical points are obvious
to everyone who understands basic mathematics - as you also
say. Forget about >>sides<< and all that personal stuff.
There is the mathematics - answer to it in mathematical terms
and with the same clarity. That would lead to a real discussion
and eventually to a clear result.

Why for example didn't we finish the discussion about object
rings. We obtained a result. We could have stated it clearly.
We would not need to start all over again discussing statements
like >>no non-unit member of the ring is a factor of any two
integers that are coprime in the ring of integers<<.
It has been stated over and over again that if the ring R you
are considering here contains the integers, then this statement
is satisfied for every ring! It is nothing particular!

Check this out: coprime integers generate the whole of Z (integers)
as an ideal. Lifting this ideal to the ring R leads to
the whole ring too. Thus the two numbers cannot be divided by
a common non-unit of R.

This is exhausting - one feels like a hamster running in his
wheel without ever moving one centimeter ahead.

H

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