
Re: can someone point me to the proof that
Posted:
Jul 23, 2013 3:15 PM


dullrich@sprynet.com wrote in news:ppiqu8d1micop548h956stpcfgoopcufjv@ 4ax.com: > 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))/(XY) and G(X,Y)= (g(X)g(Y))/(XY).
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 noncommutative rings too. But certainly it works over integral domains.
B.

