On Sunday, July 21, 2013 5:48:30 PM UTC-5, Leon Aigret wrote: > On Mon, 22 Jul 2013 00:40:51 +0200, Leon Aigret > > <email@example.com> wrote: > > > > >On Sun, 21 Jul 2013 14:24:16 -0700 (PDT), firstname.lastname@example.org wrote: > > > > > >>can someone point me to the proof that polynomials over infinite fields can be uniquely mapped to polynomial functions > > >> > > >>i.e., no two polynomials are the same function over finite fields > > > > > >The difference of the polynomials would be a non-trivial polynomial > > >with infinitely many roots. Proving that a polynomial of degree n can > > >have at most n roots is not that hard. > > > > Except for n = 0 of course.
The zero polynomial does not have degree 0 (it either has no degree, or it has degree strictly smaller than 0; otherwise, theorems about the degree of the product and other results do not work out). You will sometimes see the degree of the zero polynomial defined as -oo (minus infinity), or -1; from the point of view of Euclidean domains, it simply has no degree, as the degree function is only defined on the nonzero elements of the polynomial ring.
So the result is also true for n=0, since a polynomial of degree zero must be constant and nonzero, and hence has 0 roots.