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James Harris wrote:
> The simplest example is still the one that covers the major issue as
> consider evens as a ring and then 2 is coprime to 6, because 3 is NOT
> even.
>
> You can find a problem very similar to that with the ring of algebraic
> integers...
How is this a problem?
> My point is that certain numbers are excluded from the ring of
> algebraic integers just because they cannot be roots of monic
> polynomials with integer coefficients, just like 3 is excluded from
> evens because it's not even.
Yes, if a number does not satisfy the definition of an algebraic
integer, it is excluded from the algebraic integers. Profound, isn't
it?
> Past mathematicians never realized that all exclusion from the ring of
> algebraic integers means is that a number is not the root of some monic
> polynomial with integer coefficients--and nothing else.
Actually, I think it safe to assume that mathematicians are well aware
that exclusion from the ring of algebraic integers means that a number
is not an algebraic integer, which is exactly what the above statement
says.
> Because they stepped too far--beyond what mathematical logic
> supports--they came up with "proofs" that are not mathematical proofs,
> which ultimately just prove--that a particular number is not the root
> of some monic polynomial with integer coefficients.
Perhaps you can cite an example of such a "proof" which is not a
mathematical proof?
> BUT because they didn't realize their error and a lot of mathematical
> ideas were built on top of the error,
WHAT ERROR? I am not trying to be rude or a smartass, I genuinely have
no idea what existing belief about the algebraic integers you think is
wrong. If you are going to repeatedly post to newsgroups claiming that
there is this big problem with conventional thought on the algebraic
integers, you should reasonably expect to be asked which theorem about
the algebraic integers is untrue. You have repeatedly been asked this
question, and repeatedly ignored it. Here it is again:
Which theorem about the algebraic integers is untrue?
-Rotwang
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