- #1
kritimehrotra
- 5
- 0
Hi,
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
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