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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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Registered: 12/4/04
Re: can someone point me to the proof that
Posted: Jul 21, 2013 6:47 PM
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On Sunday, July 21, 2013 4:24:16 PM UTC-5, 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.

Arturo Magidin

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