Re: Possible flaw in the polynomial ring A[X] construction

In article <8a0b6eb1-10ba-477d-9b32-b1f048978578@xxxxxxxxxxxxxxxxxxxxxxxxxxx> "Tim Golden BandTech.com" <tttpppggg@xxxxxxxxx> writes:
....
Is it so hard to believe that in the accumulation of the subject that
errors have crept in? Let's face it, each author does struggle over
their presentation of this subject. Can you imagine being drawn down
to the level of having to define X as a "variable" or "symbol"? Those
are literal quotes within the definition, not mine. These authors are
now forced to take the topic at hand as a digest rather than diverge
over to their own specialized form. Reform at this level may not be
possible. This then leaves such a reformist author an outcast.

You are pretty flowery. But when X is a variable, the things you are
defining are polynomial functions, not the polynomials themselves. Still
do not comprehend the difference?

I am still thinking about how the reals can be claimed a subset of the
complex plane. This argument can be built many ways, with the one
simplest default being what we seem to have been discussing. Yet any
line that I draw through the complex plane can be labelled a real
line.

Yes, it can, but in that case the points on that "real line" do not form
a subring of the complex numbers.


We would then have a function mapping from a real to a complex
value. However under this thinking the product of a real x with a
complex z
x z
should go through the transform
( f(x) )( z )
where f() provides the transform from a real value to a complex value.
This would then bring the ring products into conflict since the
product on this real subset of the complex plane will not generally
match the transformed product in the complex plane.

Right. And it is different when we do the canonical mapping.

I can see that given the simple mapping that this alternate could be
dismissed as irrelevant. Yet this is the more general way to map the
reals into the complex plane and in hindsight the assumption of the
first simplistic translation is overly specific.

Wrong. The way the complex numbers are *defined* in mathematics makes
the standard embedding natural.

It was strictly
chosen because it happens to align the product and sum operations so
as to not require the transform. Nonetheless the conversion of set R
to a set C is properly taken through a function.

Not a function, an isomorphism.

Having mapped the real line onto the complex plane, then removing that
real line we are left with an open crack through the old plane; at
least this is one interpretation. I see several interpretations here
but set theory as it has been constructed declares that its elements
be unique. This type of set theory may be regarded as flawed since for
instance if I were to declare a variable of a set and a constant of
that set they may never assume the same value since there is only one
such value.

Makes no sense. If you take an element e from a set S, and create a
variable x whose values can be elements of the set S, x can get the
value e. There is nothing wrong with that, because x itself is not an
element of S. So the remainder of your consideration is complete
nonsense.

I am not a set theory expert

Right, neither an algebra expert, I would think that you are not an expert
in any field of mathematics.
--
*** t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

Quantcast