Polynomials and function space over fields

[kritimehrotra]

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

 

[matt grime]

What do you mean by 'the' function? You simply mean the inclusion right? Well, for the first case, if I take a polynomial and consider it as a function, how can it be the zero function (i.e. send every element of F to zero)? Only if it was the zero polynomial.

The second is also easy. Forget fields. Suppose I tell you, over the reals, that f(0)=1, f(1)=10 and f(3)=4, write down a polynomial through the points (0,1), (1,10) amd (3,4), you'd be able to do that easily. Well, the finite field question you ask is no harder.

 

[kritimehrotra]

Yes, by "the" function, I meant the natural function mapping each polynomial in F[x] to it's associated function in F^F.

For the first case, what you mentioned is true. I had figured that much myself, but that proves that F is infinite => ker E = 0 <=> E is 1-to-1. How do we go in the opposite direction? I.e., why is it that ker E = 0 => F is infinite?

Similarly, for the second part, what you said was the basic concept I understood. But why does this require a finite field then? Given any function in F^F, couldn't I find some polynomial that could interpolate it?

 

[matt grime]

If a field is finite it has a characteristic. It is easy to show that (x+1)^p=x^p+1 when p is the characteristic of a finite field. This is more than enough of a prod to help you with the problems you're having. If that isn't enough, then does the word Frobenious help?

 

[Kummer]

Here is a little something to add to what matt-grime was lecturing about.

Theorem: Let $$|F|= \infty$$ and $$f(x)$$ be a polynomial if $$f(\alpha)=0$$ for every $$\alpha \in F$$ then $$f(x)=0$$.

Proof: Let $$\deg f(x) = n$$ (assuming $$f(x)\not = 0$$) then $$f(x)$$ can have at most $$n$$ zeros. But it clearly does not for any $$\alpha \in F$$ is a zero. So $$f(x)$$ must be the trivial polynomial.

Note, if $$F$$ is finite and $$\deg f(x) > |F|$$ then the same conclusion can be drawn. But it need not to be always true. Consider $$F$$ a field of order 3. And $$f(x)=x^3-x$$ it maps all elements into zero.

 

[mathwonk]

the proof for surjectivity matt gave also proves non injectivity in the finite case, [besides that it is obvious, since the domain is infinite and the target is finite].