Re: Understanding the quotient ring nomenclature

Tim BandTech.com wrote:
On Jun 18, 1:14 am, Brian Chandler <imaginator...@xxxxxxxxxxxxx>
wrote:
Tim BandTech.com wrote:

Aside: Is English your first language?
Yes.

Hmm. Then please stop saying things like "definitions are offended",
because they are meaningless. (Not mistranslations as I thought they
might have been.)

You still want your polynomial to be elemental.

I think I have cracked it. By "elemental" you mean this:

[We] want the polynomial to be an element of the set of things we are
talking about.

Elements within the ring terminology are members of the domain under
consideration.

Uh oh. And if the "domain" (bad word) is the "domain" of polynomials,
are the elements not polynomials?

<skip more confusion: your arguments are not wrong enough to be
refuted>

Yes, of course we do, otherwise we wouldn't be able to talk about
them. Do you think certain patterns should be ineligible for
investigation? Consider the arithmetic of (indefinitely long, but
finite) bit strings under the operations of XOR and MUL(tiply). Do you
think these form a mathematical structure? Do you think investigation
of this mathematical structure is valid (should be allowed)? Do you
see why I choose this particular example?

If you provide accurate sum and product operations on the "bit string"
that are consistent with the ring terminology then they would form a
ring.

Oh, really? Well whaddya think "XOR" and "MUL(tiply)" are, if not well-
defined operations on bit strings? Write x^y for x XOR y, and p * q
for p MUL q, and check against the ring axioms, which you can read on
Wikipedia. Here's a start:

0 = bit string ...0000000
1 = bit string ...0000001
Check: ^ is commutative: p^q = q^p (well-known programming fact)
Check: * is associative: (a * b) * c = a * (b * c)
etc.
Check distributive law: (a ^ b) * c = (a^c) * (b^c) ... bit harder so
just choose values
a=10111011 b=101100 c=1011 (or simpler!) and calculate test case by
hand

Now for the big question: this is (take it from me, or prove it to
yourself, as you wish) a ring. The "things" to which we are applying
the ring operations are the bit strings. Question: the ring is a set,
and the elements of this set are:

(a) the bit strings
(b) 0 and 1
(c) the natural numbers
(d) polynomials over GF2
(e) hydrogen and stupidity

Take your time. Pick one (1) answer only to be the Correct Answer.

As I said to Leland I'm getting tired of this topic. As far as I can
tell you all have serious mimicry issues.

"Mimicry"? You mean you think we all copy each other? Or at least that
we all have secret meetings to make sure we all say things you are
sure are wrong? Hmm.

Brian Chandler
.

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