Question

Let F be an infinite field. Show that for two polynomials f(X), 9(X) E we have that f(X) = g(X) if and only if f(t) = g(t) for all t e F. Is this still true if F is a finite field? (Consider the polynomials X and X2 on the field Z/2).

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Answer 1

1) Let let F be an infinite field. F(X7, 9(*) EF(x)... E F(X). forms a Fine h (x) by hix) = f(x)- g(x) Then the vet I = {teF: h(t) =0} Anarimat subset of Fi. suppose : that f (x) = g(x). Therefore h(x) = a (a zero element in F(x]) => h(t)=0, xtEF = f(t) – g(t) = 0; # ĆEF 7. f(t) = gitl, Htet conversiy suppose that f(t) = g(t), HtiEFi ...! Hence I = F = {terih(t) = 0% te.. h(x! has. infinitely many roots in F... which is possible only if h(x) = 0 (a 3er ellment in F(x]). = f (x) - 9(x) = 0 = f(x) = g(x).

2) If F is finite then above result need not hold. For example take F = Z2 f101) = x and g(x) = x? Then flo) = 0 = 02 = glo) and f(1) = 1 = 12 = 90 ..flt) = g(t) tte Z But f (x) = x + x2 = g(x).