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David Bernier wrote:
> jstevh@xxxxxxx wrote:
>
> [...]
>
> > What if here you betrayed humanity and condemned it?
>
> What if Gaussian integers fail "full coverage"?
>
> David Bernier
Good question.
They don't.
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 where you get a "coprime" result where algebraically it's
meaningless, but you can't with the gaussian integers.
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.
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.
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.
BUT because they didn't realize their error and a lot of mathematical
ideas were built on top of the error, all those ideas fail, which is a
political thing as well, so just like George W. Bush and his cronies
will not admit error, neither will Barry Mazur and his.
James Harris
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