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Can someone explain why the following is true? It seems to be an "accepted fact" everywhere I search, and I can't tell why.

Let F be a field. Let E be the function from F[x] to F^F, where F[x] is the set of all polynomials over F, and F^F is the set of all functions from F to F.

Then E is 1-to-1 if and only if F is infinite, and E is onto if and only if F is finite.

Why is this true?

For the first part, I can see that 1-to-1 would mean that the kernel of E is 0. So if F is finite, there should be some non-zero polynomial in F[x] which maps to the zero function in F^F.

For the second part, I can see that if F is finite, then F^F is finite. F[x] is of course infinite, and so it makes sense that E is onto, but can someone give a more conceptual reason why? Or is that the essence of the reasoning?

Thank you!

Kriti

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# Polynomials and function space over fields

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