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

On Jul 2, 10:29 am, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <e3e27ae1-b274-43dd-b5cd-41a6715a4...@xxxxxxxxxxxxxxxxxxxxxxxxxxx> "Tim Golden BandTech.com" <tttppp...@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.

You are widening out the discussion here and I appreciate that. I am
free to compose purely from the ring of reals:
x - x^3 / 3! + x^5 / 5! - x^7 / 7! + ...
These are a series of elements
{ x,+1!,x,x,x,-3!,x,x,x,x,x,+5!,x,x,x,x,x,x,x,-7!, ... }.
The redundancy in x can be eliminated if you wish.
These elements do form a useful function when combined properly with
ring operators.
This function is named sin(x).
You say:
"the ring elements are not functions."
but your distinction is not so strong given that a composition of
elements with sum and product operators does form a function. I simply
submit that the distinction of there being a function here is a fine
attempt at covering some new ground, but that it does not yield any
new results in terms of the discussion.
Particularly we can simply consider the element
x
and call it a function, though it's not very exciting. Likewise since
you have already stated that
x * x
is an element contradicting your statement above with a function of
your own claim as one element.
These are merely the qualities of the definition of ring which we
discuss. The polynomial form happens to be one compositional pattern
of the freedoms that exist. Using your previous context we can accept
any polynomial expression as an element, though I have argued that the
distinction ought to go more as you have put it here so that
x * x
is a product of two elements, that element being assignable to say y,
but until we form that expression the proper issuance of a new element
has not actually been made. So that writing
y = x * x
then formalizes the interpretation of a single element which was being
discussed without a name. This is just quibbling and I don't see that
much is here to make any provable point on the qualities of X in the
polynomial A[X].


> 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).

Well, this is fine. We can see these as polynomials too. We have come
down to real x and we see real valued results and it is such a relief
from attempting to address the unknowable X in A[X] which has been the
focus of this thread, which cannot be real valued according to posters
on this thread. Some claim that what we are discussing here is not a
polynomial at all, but is some other branch of math called the power
series. If it's practical it seems they don't want to have anything to
do with it.


> 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.

Errr... as I understand it X is being interpreted as
Z is in Q is in R is in C ... is in X .
Oh, I see that notation is bad It's more like
Z is in Q is in R is in C ... is in A[X]
and so the subring feature is inherently built in when A is chosen as
one of those subsets.
There is a little bit of set theory here I need to review.
Like, if
C - R
is the set of complex numbers with its subset of real numbers removed
does this just leave I the imaginary numbers? Or alternatively is the
R which is a subset of C just the values
x + 0 i
where x is in R?
I believe that this is the correct interpretation, which leaves
x + 1.2 i
in the set C after R is removed, otherwise we could freely substitute
this expression into the products and sums as R, which is clearly
going to break things. I'm sorry if this digression is not easy to
follow. This could be important to the interpretation of A[X] with A
removed. This would be my misunderstanding wearing different clothes.
Copies of A will still exist in A[X] even when we pull A out of A[X].
This is consistent with my own arguments earlier.


> 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?

Please see criticisms above where I see a conflict of interpretation
in your thinking here. I know these things are extremely simple that
we are discussing, and so frustrating inherently. In that sin(x) is a
composition of elements then there is an element which can represent
sin(x) if we take the composition as an element. We are breaking down
to variables versus constants I believe, which could also be helpful
new ground. Upon establishing X as a variable then the ring definition
still demands its compositions be in the ring. Whether we label that c
(5) or sin doesn't matter much. 1, 2, and 12 are radix ten
representations above right? If not then I see only two elements with
redundancy and notation. Thanks, ***.

- Tim


> 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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