It is well know that a polynomial doesn't determine uniquely the polynomial function associated with it; e.g. $f_1(x)=x$ and $f_2(x)=x^p$ with coefficients in the finite field with $p$ elements both induce the same polynomial function, since $x\equiv x^p\ (mod\ p)$ by little Fermat's Thm. My question is: can this happen also in a field with infinitely many elements? If not, (as i suppose), why?
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1A field with infinite elements? or a field with inifinitely many elements? – Hagen von Eitzen Feb 15 '13 at 16:02
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infinitely many – Federica Maggioni Feb 15 '13 at 16:04
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If $f,g\in F[X]$ are two different polynomials such that $f(a)=g(a)$ for all $a\in F$, then $p:=f-g$ is a nonzero polynomial of some degree $d$ such that $p(a)=0$ for all $a\in F$. Thus if $a_0,a_2,\ldots, a_d$ are $d+1$ distinct elements of $F$, we find that $p$ is a multiple of $(X-a_0)\ldots(X-a_d)$, a polynomial of degree $d+1$, which is absurd.
Hagen von Eitzen
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1did you mean: $a_0,a_1,\ldots a_d$ are $d+1$ distinct elements....? – Federica Maggioni Feb 15 '13 at 16:11
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