On Sunday, July 21, 2013 4:24:16 PM UTC-5, 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 f-g 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 f-g 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^k-x 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 one-to-one if and only if F is infinite.