Re: Congruence question.
- From: Tonico <Tonicopm@xxxxxxxxx>
- Date: Thu, 12 Jun 2008 23:31:31 -0700 (PDT)
On Jun 13, 5:43 am, Bill Dubuque <w...@xxxxxxxxxxxxxxxxxxxx> wrote:
r...@xxxxxxxxxxxxxx (Rob Johnson) wrote:
In article <y8zhcbyz2pj....@xxxxxxxxxxxxxxxxxxxx>,
Bill Dubuque <w...@xxxxxxxxxxxxxxxxxxxx> wrote:
Saysero <says...@xxxxxxxxx> writes:
Prove that for any prime p, if a^p - b^p is divisible by p
then a^p - b^p is also divisible by p^2.
n n
x - y n-1 n
HINT In any ring ------- = n x = D (x ) if x = y
x - y
See also my prior posts [1] on the algebraic derivative for polynomials
[1]http://google.com/groups?selm=y8z64ry3745.fsf%40nestle.csail.mit.edu
http://google.com/groups?selm=y8zr6b6fd92.fsf%40nestle.csail.mit.edu
If we allow division by x - y when x = y, then we allow the following
common fallacy:
Assume x = y, then
2
xy = y
2 2 2
x - xy = x - y
2 2
x(x-y) x - y
------ = -----
x-y x - y
x = 2x
Then either all x = 0, or 1 = 2. Both of these restrict the ring
to one element.
The algebraic formalism you cite above works if we carry out the
division first, without assuming x = y, then assume x = y in the
resulting quotient. However, if we let x = y before the division,
problems can arise. In most rings, 0/0 is reasonably undefined.
You've completely misunderstood - please read the cited posts.
--Bill Dubuque-
*********************************************************
Every time, or at least of times, you post something, it looks pretty
odd, pretty bizarre, and even wrong. You then send everybody to "read
your posts on this or that", but that usually helps little, since the
info is hidden most of the times in links to other posts which have
links to other posts, etc.
In the last discussion about determinants, I followed your posts and I
went back to 1999 in one of those paths, so I stopped since I have no
pro training in archeology.
Other links take us to posts with links to posts with links to posts
...to posts where it all stops! Then we learn that you haven't yet
developed some thing, so it all stops and we get nothing.
I think the above situation is pretty confusing, so I insist: if you
think you have some really good "general principles" in general
algebra that are appliable to a wide range of problems, why won't you
please write them donw in some site so that people will be able to
read them concentrated in one single place?
I know this may be a burden to you, but (1) I, at least, would thank
you very much if you'd do it, and (2) these discussions about "this is
not true", "yes, it is: read on" and etc. will stop, and you'll be
able to make your point crystal clear every time.
Regards
Tonio
.
- Follow-Ups:
- Re: Congruence question.
- From: Michael Press
- Re: Congruence question.
- From: Bill Dubuque
- Re: Congruence question.
- References:
- Congruence question.
- From: Saysero
- Re: Congruence question.
- From: Bill Dubuque
- Re: Congruence question.
- From: Rob Johnson
- Re: Congruence question.
- From: Bill Dubuque
- Congruence question.
- Prev by Date: Re: Primes of the form 2^(2^n)+1
- Next by Date: Question about representations of Lie algebra sl(n).
- Previous by thread: Re: Congruence question.
- Next by thread: Re: Congruence question.
- Index(es):
Relevant Pages
|
