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Topic: can someone point me to the proof that
Replies: 9   Last Post: Jul 24, 2013 10:39 AM

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 David C. Ullrich Posts: 3,555 Registered: 12/13/04
Re: can someone point me to the proof that
Posted: Jul 22, 2013 11:15 AM

On 21 Jul 2013 21:43:22 GMT, Bart Goddard <goddardbe@netscape.net>
wrote:

>lax.clarke@gmail.com wrote in news:f6288bc3-b3b4-45bb-9425-2a70f2cea066
>

>> no two polynomials are the same function over finite fields
>
>I think you mean _infinite_ fields. In which case you can
>use functional methods. If the two polynomials give the
>same function, then plug 0 in for X to see that their constant
>terms are the same. Then take (formal) derivatives and plug
>in 0 again to see that the first-order coefficients are the
>same, etc.

Not that it matters, since the result is easy to prove
otherwise, but I don't see how this works. The problem
being that the formal derivative is just "formal":

Looking at the difference of our two polynomials,
say p(t) = 0 for all t in our infinite field. So p
has zero constant term (hence p(t) = t q(t) for
some polynomial t and we're done, hence the
"not that it matters" above). How does
it follow that p'(t) = 0?

That certainly follows for real or complex polynomials,
since the formal derivative is also the derivative,
with a definition in terms of the _values_ of
the asspciated polynomial function. But..

>
>B.

Date Subject Author
7/21/13 Lax
7/21/13 Bart Goddard
7/21/13 Lax
7/22/13 David C. Ullrich
7/23/13 Bart Goddard
7/24/13 David C. Ullrich
7/21/13 Leon Aigret
7/21/13 Leon Aigret
7/21/13 magidin@math.berkeley.edu
7/21/13 magidin@math.berkeley.edu