(a) Let $f \in \mathbb { Z } [ x ]$ be a non-constant polynomial with integer coefficients. Show that as $a$ varies over the integers, the set of divisors of $f ( a )$ includes infinitely many different primes. (b) Assume known the following result: If $G$ is a finite group of order $n$ such that for integer $d > 0$, $d \mid n$, there is no more than one subgroup of $G$ of order $d$, then $G$ is cyclic. Using this (or otherwise) prove that the multiplicative group of units in any finite field is cyclic.
(a) Let $f \in \mathbb { Z } [ x ]$ be a non-constant polynomial with integer coefficients. Show that as $a$ varies over the integers, the set of divisors of $f ( a )$ includes infinitely many different primes.\\
(b) Assume known the following result: If $G$ is a finite group of order $n$ such that for integer $d > 0$, $d \mid n$, there is no more than one subgroup of $G$ of order $d$, then $G$ is cyclic. Using this (or otherwise) prove that the multiplicative group of units in any finite field is cyclic.