[solved]Need a proof of the polynomial generalization of fermat's little theorem
Disclaimer: I'm not a mathematician, nor a student of math. So my knowledge isn't really good and I'm not accustomed at proving theorems.
I need a short and simple proof of this generalization of Fermat's littel theorem:
Theorem: Let and such that and are coprimes.
Then is prime if and only if (eq).
I found a partial proof in one of my books, but there is some part left "as an exercise" which I don't understand how to prove.
The partial proof looks like this:
The coefficient in of the term, for is , and if is prime
then . Thus, for Fermat's little theorem, the (eq) holds(*).
Let us suppose is a composite number and let be one of the prime factors of . Let be the maximum integer such that ,
then (**) and it is relatively prime with .
Thus the coefficient of is non-zero modulo , therefore is not the null-polynomial in ,
or, in other words, (eq) does not hold.
(*) Not complete, but It's trivial to prove. Obviously because the at the numerator is not simplified. Applying the fermat's little theorem to the remaining terms(=first and last) yields the result.
(**) This is the statement left as an exercise, and which I want to prove.
I know that the proof of that little statement must be simple, but I can't find a way of proving it. I'd like to find some very small and simple proof using combinatorics.
Anyone can help me in this task?
By the way: before posting this thread I've searched in this forum, on google, on mathoverflow, but I couldn't find a proof.
I found some references of books containing the proof, but, by what I understood, they prove a more general theorem for
finite fields, so probably the proof requires a deeper knowledge of field theory, and also I haven't got these books at hand.
If anyone knows also the finite-field version, it's welcome to post it.
Re: Need a proof of the polynomial generalization of fermat's little theorem
Okay, I think I came up with a proof.
Suppose , then it means that .
This means that:
Or, written in an other way:
Now, for , since is prime, and are coprimes if and only if .
And that's true because, we supposed that , so if we would have , but , so that's absurd.
This imply that in the previous equation, the right hand side is an integer , but the denominator does not divide ,
which means that the denominator must divide the rest of the numerator, but this imply that , which is absurd because we supposed
to be the biggest exponent such that .