
Cyclotomic Polynomials 253
Recall (Exercise 21.5) the following property of roots of unity:
1 + θ
j
+ θ
2 j
+ ···+ θ
(p−2) j
=
p −1 if j = 0
0 if 1 ≤ j ≤ p −2
Therefore, from (21.8),
ζ =
1
p−1
[α
0
+ α
1
+ ···+ α
p−2
]
=
1
p−1
[
p−1
p
β
0
+
p−1
p
β
1
+ ···+
p−1
p
β
p−2
]
(21.9)
which expresses ζ by radicals over Q(θ).
Now, θ is a primitive (p−1)th root of unity, so by induction θ is a radical expres-
sion over Q of maximum radical degree ≤ p −2. Each β
l
is also a radical expression
over Q of maximum radical degree ≤ p −2, since β
l
is a polynomial in θ with ra-
tional coefficients. (Actually we can say more: if p > 2 then p −1 is even, so the
maximum radical degree is max(2,(p −1)/2). Note ...