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Galois Theory, 4th Edition
book

Galois Theory, 4th Edition

by Ian Nicholas Stewart
March 2015
Intermediate to advanced content levelIntermediate to advanced
344 pages
10h 18m
English
Chapman and Hall/CRC
Content preview from Galois Theory, 4th Edition
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
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Publisher Resources

ISBN: 9781482245837