Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: can someone point me to the proof that
Replies: 9   Last Post: Jul 24, 2013 10:39 AM

 Messages: [ Previous | Next ]
 magidin@math.berkeley.edu Posts: 11,749 Registered: 12/4/04
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 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.

--
Arturo Magidin

Date Subject Author
7/21/13 Lax
7/21/13 Bart Goddard
7/21/13 Lax
7/22/13 David C. Ullrich
7/23/13 Bart Goddard
7/24/13 David C. Ullrich
7/21/13 Leon Aigret
7/21/13 Leon Aigret
7/21/13 magidin@math.berkeley.edu
7/21/13 magidin@math.berkeley.edu