Question

Let F be a field and f(x), g(x) ? F[x] both be of degree ? n....

Let F be a field and f(x), g(x) ? F[x] both be of degree ? n. Suppose that there are distinct elements c0, c1, c2, · · · , cn ? F such that f(ci) = g(ci) for each i. Prove that f(x) = g(x) in F[x].

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Answer #1

Proof:

Let . Suppose   then .  

Let   be elements in such that .

Then  

are zeros of  .

Hence ,   has   zeros which is not possible.

must equal  .

Hence  , .

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