Re: Is the polynomial ring interpretation of abstract algebra A[X]

In article <e3e27ae1-b274-43dd-b5cd-41a6715a4d5e@xxxxxxxxxxxxxxxxxxxxxxxxxxx> "Tim Golden BandTech.com" <tttpppggg@xxxxxxxxx> writes:
On Jul 1, 10:21 am, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <d49f93ae-4ad7-4a62-a196-b5b1b8eee...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> "Tim BandTech.com" <tttppp...@xxxxxxxxx> writes:
...
> The only substantial disagreement that I see is that you claim that
> for a polynomial of real coefficients that those coefficients aren't
> actually real.

That was not the claim. In a polynomial ring over some base ring A the
"coefficients" are from the ring A. Moreover, the set A is a subset of
A[X], i.e. each element of A is also an element of A[X].

> We are not free to mix products as
> x z
> where x is real and z is complex since these are two sets,k not one.

We are, because R is a subring of C.

> Your own looseness here is exactly part of the problem. As I
> understand it a specification of
> polynomial with real coefficients

You are again confusing the polynomials as a function and as the elements
of a polynomial ring. Meaning that you still do not understand what a
polynomial ring actually is.

I don't see that the functional distinction does anything for the
conflict that I am expressing.

It does show that there is no conflict.

In functional systems there is a domain and a range. Under the ring
nomenclature these are the same, so long as you work in products and
sums.

May be. But in a polynomial ring the ring elements are not functions.
Addition and multiplication is defined between the elemens of that ring,
and those operations make it a ring. It is called a polynomial ring
because the elements can be seen as polynomials with coefficients of
a base ring.

The function does not distinguish the conflict and furthermore
does allow for real valued functions such as sin(x) to be expressed
via the polynomial, with some additional constraints.

May be, but we can also define a "functional ring" where the elements are
functions of a base ring, as addition and multiplication are defined, this
also forms a ring. (But it is better to use only functions that are
defined everywhere on the base ring.) So with R the base ring, sin(x)
and cos(x) are elements of that functional ring and sin(x)*cos(x) is
another element, as is sin(x)+cos(x).

I wonder if given your statement that R is a subring of C if the same
can be proven of X.

What about X? X is not a ring, so it can not be a subring of something,
nor can something be a subring of it.

If it is not flawed than at least it grants X some
qualities beyond a symbol. Beneath here I see you are claiming that
X X
is a new element and I see this yet this new element is composed of
two elements. Until we mark this new element
c = X X
then we do not have an element.

Why not? What is in a name? Why would c be a name such that we have a
new element and X^2 not? Why are 1, 2 and 12 three different elements of
N where I see only 1 and 2?

Upon accepting the element c we then
have the collapse of the polynomial.

What collapse?

All of this garbage goes away by simply taking the ring definition as
pristine. Working in a generic ring A polynomials can be expressed and
could be applied to any specific ring, but upon specifying A as say
the reals then all products and sums become real valued.

These are polynomials seen as functions. But when seen as elements of a
separate ring is something completely different, and I do not see why you
keep thinking that that is not a ring.
--
*** t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
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