First the rabbit. The introduction of this function is the part which I don’t know how to motivate. Let be a polynomial and define
Integration by parts gives the recursion
and therefore we have the formula
Now suppose (for want of a contradition) that . Let the set of Galois conjugates of be Then we have , expand this as and rewrite as
where the are the nonzero exponents.
Let be an integer such that is an algebraic integer. Let be a (large) prime and we choose to take
There are absolute constants and (independent of ) such that
(look at the integral definition of and apply the naive estimate).
Now consider . It is a Galois-invariant algebraic integer, hence an integer. We have
Here higher order means at least derivatives appearing. Each of these higher order terms is divisible by , hence by . Since is prime, for large enough , is not divisible by , hence nonzero.
Now every term is divisible by , so we get the lower bound
As there are infinitely many primes, we can send choose large enough to get a contradiction, QED.