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

On Jul 1, 9:21 am, "Dik T. Winter" <Dik.Win...@xxxxxx> wrote:
In article <d49f93ae-4ad7-4a62-a196-b5b1b8eee...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> "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].

That actually depends on the construction used. If you define the
underlying set of A[X] as a set of formal expressions, then yes; if
you define the underlying set of A[X] as a set of almost null
sequences with values in A, then no.

However, in *all* cases, there is a subset of A[X] that can be put in
one-to-one, addition and multiplication preserving correspondence with
A; that correspondence may be the identity, or it may be a different
function. So we generally identify A with the corresponding subset of A
[X] and think of A as being contained in A[X].

If you go back through the thread, you will see the OP purposely
misunderstanding and misrepresenting at every opportunity in which
some liberties were taken with notation or with nomenclature. The
current dead horse he is abusing is precisely of that kind. At the
same time, when presented with a strict, formal construction using
sufficient symbols so that no liberties were taken, he had no option
but to beat a quick withdrawal and then claim it was never presented.

--
Arturo Magidin
.

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