
Reduction Modulo p 57
Proof. Note that f (t) = (t
p
−1)/(t −1). Put t = 1+u where u is a new indeterminate.
Then f (t) is irreducible over Q if and only if f (1 + u) is irreducible. But
f (1 + u) =
(1 + u)
p
−1
u
= u
p−1
+ ph(u)
where h is a polynomial in u over Z with constant term 1, by Lemma 3.21. By Eisen-
stein’s Criterion, Theorem 3.19, f (1 + u) is irreducible over Q. Hence f (t) is irre-
ducible over Q.
Setting p = 17 shows that the polynomial (3.3) is irreducible over Q.
3.5 Reduction Modulo p
A second trick to prove irreducibility of polynomials in Z[t] involves ‘reducing’
the polynomial modulo a prime integer p.
Recall that if n ∈ Z, two integers a