Re: Understanding the quotient ring nomenclature

Tim BandTech.com wrote:
On Jun 18, 11:50 am, Brian Chandler <imaginator...@xxxxxxxxxxxxx>
wrote:
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.)

I will stand by this language...

Suit yourself. Don't comlain that no-one knows what you are talking
about, though.

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?

No. The polynomial is expressed on the ring A.

<Etc. Skip lecture on ring theory>

Again, you have a choice. You started by claiming you wanted to
understand a quotient ring construction, but make it clear you are
only interested in telling us it's wrong [or "offended" perhaps]. If
you want to learn, you have to listen.

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.

a) the bit strings. When selecting one member of the set we will
always get a bit string.

Really? I'm sure that if I had studied "ring theory" under your
tutelage I would have claimed (b).

But actually (a) is one correct answer -- (d) is another one. Can you
do formal manipulation of polynomials over a ring other than integers
or similar? It's not difficult, and you can do it without worrying
whether you are one of Sawyer's "Bright" or "Dull" pupils.

Examples (we write polynomial answers with no spaces around the +
signs; spaced + means "add the polynomial on the left to the
polynomial on the right". After the answers I am writing the
corresponding bit string.

1. 1 + x = x+1 : 11
2. 0 + x^7 = x^7 : 10000000
3. x^3 + x^3 = 0 : 0 (because 2*anything = 0)
4. x^4+x + x^2+x+1 = x^4+x^2+1 : 10101

Got that? You do some:

5. x^4+1 + x^3+x+1 =
6. x^6+1 + x^5+x^4+x^3+1 =

Now try some multiplication:

7. (x+1)(x+1) = x^2+1 : 101
8. (x^2+1)(x) = x^3+x : 1010

Off you go:

9. (x^2+1)(x^2+1) =
10. (x+1)(x^3+1) =

Now if you _really_ want to understand anything, you will check the
bit twiddling method for the same examples, and see if it produces the
same pattern. If two things have the same pattern, in maths we're not
interested *what* they are, or what they are 'made' of (wood, glass,
metal), we identify their structures.

Here's just the last one of each of the examples I did

4. x^4+x + x^2+x+1 : 10010 ^ 111 = 10101 (xor, remember)
8. (x^2+1)(x) : 101 * 10 = 1010

Incidentally, since as you know, these bit strings also represent the
natural numbers, you could have given (c) as a correct answer,
provided you note that this is not the naturals with the "usual"
addition and multiplication.

Brian Chandler

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