Re: New paper, algebraic integers, Galois Theory
From: W. Dale Hall (mailtowd-hall_at_pacbell.net)
Date: 10/19/04
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Date: Tue, 19 Oct 2004 05:43:50 GMT
James Harris wrote:
> "W. Dale Hall" <mailtowd-hall@pacbell.net> wrote in message news:<diFcd.14951$nj.9069@newssvr13.news.prodigy.com>...
>
... stuff deleted ...
>>>>Now, your correction, in which you now state that in this situation,
>>>>which merely restates the definition of coprimeness in the particular
>>>>situation at hand:
>>>>
>>>> a_i and 5 are coprime in the ring of algebraic integers
>>>> if and only if there exist algebraic integers u and v
>>>> for which
>>>>
>>>> u a_i + v 5 = 1
>>>>
>>>>you now *NO LONGER CLAIM* to have proven that u and v exist.
>>>
>>>
>>>Well, yeah, as the assertion is equivalent to claiming that there
>>>exists an algebraic integer that is the root of a non-monic polynomial
>>>with integer coefficients, which it can't be.
>>>
>>
>>So such u and v do not exist. Right?
>>
>
>
> No, they do not exist.
>
>
Thus, you agree that the numbers a_i and 5 are not coprime in the ring
of algebraic integers, contrary to the claim of your paper.
Done.
When will you retract your paper?
... stuff deleted ...
> Let's say someone defines a ring where 3 is not a factor.
>
> Then they say that 9 is prime.
>
And again we pretend that your toy example really has
9 being prime in the ring ...
> You reply that their definition is silly, and 9 = 3(3).
>
> They say, but it's the DEFINITION!
>
> The argument can go on, and on, and on.
>
Only when one party fails to understand that primeness
is a property that depends on the ring in which you're
doing business.
If the participants are mathematicians, then they
already realize that the issue is relative. They
compare definitions (oh, DEFINITIONS?) and determine
what each other is talking about.
There is no argument.
> The ring of algebraic integers excludes numbers that are not roots of
> monic polynomials with integer coefficients.
>
> There's no need to argue here, as common ground can be found.
>
> You assert that the *definition* of coprimeness is one thing.
>
> I say, ok, but it contradicts with algebra.
>
No, unless you are saying that by merely uttering the
terms "algebraic integer" and "coprime", you invoke
a contradiction.
Why is that?
The reason that ignoring the algebraic integers, or
looking at some other domain, such as your alleged
"object ring", does NOT eliminate the set of zeros
of monic integer polynomials.
It is pure folly to assume that it does.
Similarly, despite the unseemly fretting you've
been doing over coprimeness, all it amounts to
is the existence or non-existence of a particular
combination of the ring elements you're talking
about. Eliminating the discussion of coprimeness
still leaves you with some sets of values that
can be used in certain combinations to produce 1,
while others cannot be used in those combinations
to produce 1.
So, you make some sort of contradiction. Hooray
for you. However, that *same* contradiction can
be expressed by substituting "zero of a monic
polynomial over the integers" for "algebraic
integer", and by substituting for "x,y are coprime
in the ring of algebraic integers", the phrase
"there are roots of monic integral polynomials u
and v, for which ux + vy = 1", and by substituting
for "x,y are not coprime in the ring of algebraic
integers", the phrase "for all roots of monic,
integral polynomials u and v, ux + vy != 1".
If you think about it, there is no need for the
definitions "algebraic integer" and "coprime in
the ring R". Everything that can be expressed
in terms of them can be expressed without them.
> I agree with you about your definition, saying it's arbitrary, and
> that I can prove it is and that it leads to a direct contradiction
> with algebra itself.
>
So, you're saying that you can produce a contradiction
with algebra itself, even without the two constructions,
merely by talking about "roots of monic polynomials over
the integers", and "capable of producing 1 by means of a
combination ux + vy".
Such as this:
The numbers a_i and 5 are each roots of monic polynomials
over the integers, and for any u and v, also roots of monic
polynomials over the integers, the combination
u a_i + v 5
is never equal to 1.
Apparently, this *true* statement leads to a contradiction.
Note how it does not require one to speak of the banished
terms "algebraic integer" nor of the dread "coprime".
> My argument is that algebra trumps an arbitrary definition.
>
Indeed. Except I venture to guess that you will take exception to what
I've just said. You still fail to understand the depth of your folly.
> Are you ready to objectively discuss this issue now, then?
>
This is silly. I have stayed on the point of what the actual
definitions are, and have continued to remind you of what
things actually mean, despite your determined attitude that
the definitions are wrong.
In mathematics, definitions are (by definition) not wrong.
They can be meaningless, vacuous, worthless, or end up
being definitions of nothing at all. As Arturo Magidin
and others have repeatedly said, a definition is merely
a shorthand: an abbreviation into concise terms of some
thing or concept that someone thought of, and others
continue to use.
With precious few exceptions (if any), a definition is never
indispensible. Your "contradiction" due to the use of
algebraic integers and "coprimeness" is still present, only
couched in more cumbersome language.
>
> James Harris
Dale
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