This is from Rotman's Group Theory book, although I don't have the specific reference right now, as the book is with a friend. He asks to show that $\alpha (x) = 4x^2 - 3x^7$ is a permutation of the elements of $\mathbb{Z}_{11} = \mathbb{Z} / 11\mathbb{Z}$.
I can show this by brute force calculation, but I feel there must be some more elegant way to show this. I looked at the Wikipedia page on permutation polynomials, which, based on a quick perusal, seemed to suggest that it is not an easy question when the degree of the polynomial is greater than 2. It also mentioned the Dickson polynomials, but unless I am missing something, the polynomial in the question isn't a Dickson polynomial.
I understand that there might not be a general method for this type of question, but even if someone can help me understand this particular example better, I would appreciate it. In particular, what I'd like to know is:
- is there a nice way to build a permutation polynomial given some quotient of $\mathbb{Z}$ to be permuted?
- can we say something about the inverse of the polynomial -- is it also a polynomial? How is it related to the initial polynomial?
- is there some faster way than plugging in to see that this polynomial must act as a permutation on $\mathbb{Z}_{11}$?
- what other sets $\mathbb{Z}_{n}$ will be permuted by this polynomial? How can we see this?
Answers to any or all of these questions, or explanations why there are not good answers, would be much appreciated. Thanks!
p.s. this is my first post, so please feel free to edit/re-tag as necessary as I am not yet sure how this all works.