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Topic: Understanding the quotient ring nomenclature
Replies: 423   Last Post: Aug 3, 2012 7:42 AM

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 Denis Feldmann Posts: 490 Registered: 9/12/07
Re: Understanding the quotient ring nomenclature
Posted: Jun 6, 2009 3:55 PM

Arturo Magidin a écrit :
> On Jun 6, 9:44 am, "Tim BandTech.com" <tttppp...@yahoo.com> wrote:
>

>> Thanks. It certainly does feel vague. For instance if I were to build
>> a polynomial in this X with real coefficients such as
>> - 5.2 XXX + 2.3 XX - 1.1 X - 0.23
>> I'd have rather a lot of trouble evaluating this for a given X if X is
>> undefined.

>
> You are confusing polynomial *functions* with polynomials. Polynomials
> are not "evaluated". You don't need to "evaluate for a given X". (And,
> on the other hand, if you have a polynomial *function*, then a "given
> X" would, *of course*, not be "undefined": it would be *given*).
>

>> On the other hand, if X is real
>
> X is not a real. These things are not *functions*, they are
> *polynomials*.
>

>> then regardless of what I
>> put in this expression the result is likewise real.

>
> These are not expressions to be evaluated, they are polynomials.
>

>> Somehow
>> mathematicians are comfortable working with objects which are not
>> clearly instantiated.

>
> Polynomials *are* "clearly instantiated". The problem here is not
> about what mathematicians are or are not comfortable working with, but
> rather that you are confusing two things (not your fault: in high
> school most people are not told of the distinction). Polynomials and
> polynomial functions are *different things*. See Bill Dubuque's post
> if you want a particular way of instantiating polynomials as almost
> null sequences.
>
>

>> I'm afraid that at this level of discussion I'm
>> dealing in several of these stacked uninstantiated objects.

>
> I'm afraid that the real problem is that you are a bit confused about
> the objects in question, and that there are some rather large lacunae
> in your knowledge of the things you need to understand these objects.
> Again, not your fault, but you are trying to run a marathon without
> knowing how to move your feet.
>
>

>> Above here Arturo states:
>>
>> "R is the ring of real numbers, X is an indeterminate variable, and
>> R[X] is the ring of polynomials with coefficients in R."
>>
>> yet this does not wash since there are three devices described in a
>> notation of just two devices.

>
> If by "this does not wash" you mean "I do not understand", then you
> are correct. The problem here is that you are not understading, not
> that anyone is hiding anything from you.
>
> The ring of polynomials S[T], with S a ring and T a set is
> *completely* determined by specifying S and T. The "third object" is
> *constructed* from the first two.
>
>

>> The polynomial has a clear description
>> as a sum
>> sum over n ( a(n) X ^ n )
>> yet is never actually invoked within the description.

>
> And now you are confusing a *particular* polynomial with the ring of
> *all* polynomials. R[X] is the ring of *all* polynomials with real
> coefficients, not a particular one. See? It's not that what I said
> "does not wash": it's that you are not familiar enough with these
> things to understand what is being said.
>

>> The meaning of X has been scuttled.
>
> Nonsense.
>

>> Your interpretation seems more believable but you
>> say it is not the truth.

>
> Look: nobody is lying to you. The problem here is that you are simply
> not aware of the context and the background necessary to understand
> the answers you are given, but you are now blaming those answering you
> and even accusing them of lying to you, because you don't understand
>
> Nobody is lying to you. Nobody is hiding the truth. What you need to
> do is learn a bit more about rings and polynomial rings in order to
> understand the answers you are being given.
>

>> I guess that one simple question to get answered is:
>> Is X real valued?

>
> Sigh. You are not dealing with a specific polynomial nor with
> polynomial functions. Your question may be simple, but it is in fact
> quite meaningless in this context. Just as if you were asking "what
> color are ideas that dream furiously?".
>

I know that one, I do. Can I answer, teacher? Can I ?
> --
> Arturo Magidin

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