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Some recent threads simply explaining yet another way to see the
coverage problem of the ring of algebraic integers have floundered on
the issue of convergence and what it means with rings like the ring of
algebraic integers, where I introduce the concept of an exclusionary
ring.
The concepts are simple, luckily, as I can mostly use something basic:
S = 1 + x + x^2 + x^3 +...
where the issue is convergence, so that if S converges in whatever ring
you're in--notice none given yet--you can go to
S = 1 + x*S
and solve to get
S = 1/(1-x)
where to get convergence in the ring of complex numbers, which of
course is also the field of complex numbers, let's choose x = sqrt(5) -
2, considering only the positive solution, then in complex numbers, I
have
S = 1/(1 - sqrt(5))
which is a complex number, and it's also an algebraic number, but it's
NOT an algebraic integer.
But if I go back to
S = 1 + x + x^2 + x^3 +...
and plug in x=sqrt(5) - 2, I'll have
S = 1 + (sqrt(5) - 2) + (sqrt(5) - 2)^2 + (sqrt(5) - 2)^3 +...
and if I stop at some point like just look at
1 + (sqrt(5) - 2) + (sqrt(5) - 2)^2
I have an algebraic integer, and in fact, I can have an arbitrarily
large number of terms added together, and stop--ever closer to
1/(1-sqrt(5)) and STILL have an algebraic integer.
BUT in the ring of algebraic integers, you cannot reach 1/(1-sqrt(5))
because it is NOT an algebraic integer!
Yet you can approach it out to infinity, so you have an asymptotic
approach to that value.
A corollary to this is 1/x where you can let x go out to infinity and
that approaches 0 but never reaches it, or you can let x approach 0,
but never reach it, but the difference with the infinite sum example is
that the asymptotic nature is created by the exclusionary nature of the
ring of algebraici integers!!!
That is, the reason you can never reach 1/(1-sqrt(5)) with the series
is that 1/(1-sqrt(5)) is NOT the root of any monic polynomial with
integer coefficients, which is the rule that defines algebraic integers
and excludes that value!
So the mathematics holds on a definition, as logically, if
1/(1-sqrt(5)) is not an algebraic integer, then there is no way to
reach that value in the ring, so
S = 1 + (sqrt(5) - 2) + (sqrt(5) - 2)^2 + (sqrt(5) - 2)^3 +...
approaches it asymptotically, in the ring of algebraic integers.
But now you have a problem with
S = 1 + x*S
as it's just not necessarily true in the ring of algebraic integers.
Why not necessarily true?
Because the exclusion does NOT apply if 1-x is a unit in the ring of
algebraic integers.
So if it is a unit then the series can converge within the ring, but
otherwise the exclusionary rule--the definition of algebraic integers
as roots of monic polynomials--prevents.
Now I feel confident there are posters who will want to dispute me on
the ability to use convergent infinite series in the ring of algebraic
integers, but I want more than namecalling, or other childish rants,
and I want more than someone saying they read it different in some
number theory text.
The problem here again is that people who don't know mathematics that
well, can write books.
And other people can believe things that are wrong for quite some time,
and LOTS of people can believe things that are wrong, so simply saying
a lot of people have believed other things for a long time is not the
way to reply back to me here.
If any of you actually know mathematics versus being people who can
repeat what you're told as if that's all that matters, then you can
explain the status quo view.
You can explain the mainstream beliefs of the mathematical community in
this area without relying on insults, without simply claiming that I'm
wrong, and without doing anything other than objectively replying.
I know that can't be done, so I say that upfront to remind readers that
a lot of people around the world are fighting the truth with my
research to preserve their delusion of expertise, which is all about a
lot of people getting some mathematics wrong.
James Harris
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