
258 Circle Division
which is a direct consequence of their definition. We can use this identity recursively
to compute Φ
n
(t). Thus
Φ
1
(t) = t −1
so
t
2
−1 = Φ
2
(t)Φ
1
(t)
which implies that
Φ
2
(t) =
t
2
−1
Φ
1
(t)
=
t
2
−1
t −1
= t + 1
Similarly
Φ
3
(t) =
t
3
−1
t −1
= t
2
+t + 1
and
Φ
4
(t) =
t
4
−1
(t −1)(t + 1)
= t
2
+ 1
and so on. Table 21.8 shows the first 15 cyclotomic polynomials, computed in this
manner. A curiosity of the table is that the coefficients of Φ
n
always seem to be 0,1,
or −1. Is this always true? See Exercise 21.11.
n Φ
n
(t)
1 t −1
2 t + 1
3 t
2
+t + 1
4 t
2
+ 1
5 t
4
+t
3
+t
2
+t + 1
6 t
2
−t + 1
7 t
6
+t
5
+t
4
+t
3
+t
2
+t + 1
8 t
4
+ 1
9 t
6
+t
3
+ 1
10 t
4
−t
3
+t
2
−t + 1
11 t
10
+t
9
+t
8
+t
7
+t
6
+t
5
+t
4
+t
3
+t
2
+t + 1
12 t
4