1. Let G be a finite group. Assume for any d ∈ N dividing the order of G we have
|{g ∈ G | g d = e}| ≤ d.
Show that G is cyclic (notice that we do not assume that G is abelian, so you cannot
use
P
classification of finite abelian groups as in the book). Hint: use the formula n = d|n ϕ(d)
and its proof.
2. Let f ∈ Z[x], deg f ≥ 1. Denote by P the set of prime numbers. Let S ⊂ P be the
subset of primes p such that f mod p has a root in Zp . Equivalently, p ∈ S if there exists
n ∈ Z such that p |f (n). Show that S is infinite. In particular, even if f is irreducible in
Q[x], its reduction modulo p is reducible in Zp [x] for infinitely many p’s.
In fact, much more is true: S has positive lower density in P in the sense that
lim inf
n→∞
|{p ∈ S | p ≤ n}|
> 0.
|{p ∈ P | p ≤ n}|