Roots of polynomials and the derangement problem

<p>We present a new killing-a-fly-with-a-sledgehammer proof of one of the oldest results in probability which says that the probability that a random permutation on&nbsp;<em>n</em>&nbsp;elements has no fixed points tends to e-1 as <em>n</em>&nbsp;tends to infinity. Our proof stems from the connection between permutations and polynomials over finite fields and is based on an independence argument, which is trivial in the polynomial world.</p>