
Re: can someone point me to the proof that
Posted:
Jul 21, 2013 6:47 PM


On Sunday, July 21, 2013 4:24:16 PM UTC5, lax.c...@gmail.com 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
Let f and g be two polynomials; the polynomial fg is either the zero polynomial, or else it has a degree. If it is of degree n, then it has at most n roots; thus, if f and g are polynomials and they agree on infinitely many elements of the field when viewed as functions, then fg has infinitely many roots in the field, and therefore must be the zero polynomial, so f=g. Thus, if f and g are polynomials over F, and they agree as function son infinitely many elements of F, then f=g.
Conversely, if F is a finite field with k elements, then x^kx evaluates to 0 on all elements of F, but is not the zero polynomial.
Thus, the canonical map form F[x] to F^F is onetoone if and only if F is infinite.
 Arturo Magidin

