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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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Bart Goddard

Posts: 1,706
Registered: 12/6/04
Re: can someone point me to the proof that
Posted: Jul 23, 2013 3:15 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply wrote in news:ppiqu8d1micop548h956stpcfgoopcufjv@

> 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?

If the two polynomials are f(X) and g(X), then let
F(X,Y) = (f(X)-f(Y))/(X-Y) and G(X,Y)= (g(X)-g(Y))/(X-Y).

Since f and g are equal for all values of X, so are F and G
for all values of X and Y. Since f'(X) = F(X,X) and
g'(X)=G(X,X), it follows that f'=g'. I think we can make
this work in non-commutative rings too. But certainly
it works over integral domains.


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