Math 5126
Monday, March 31
Ninth Homework Solutions
1. (a) Let
ω
be a primitive
p
th root of 1. Then the minimal polynomial of
ω
is
x
p

1
+
x
p

2
+
···
+
x
+
1; in particular it has degree
p

1, and the primitive
p
th roots
of 1 are precisely
ω
r
where 1
≤
r
≤
p

1. Suppose we have a linear relation
a
1
ω
+
···
+
a
p

1
ω
p

1
where
a
i
∈
Q
for all
i
. Since
ω
6
=
0, we see that
a
1
+
a
2
ω
+
···
+
a
p

1
ω
p

2
=
0
.
This shows that
ω
satisﬁes the polynomial
a
1
+
a
2
x
+
···
+
a
p

2
x
p

3
+
a
p

1
x
p

2
,
which has smaller degree than the minimal polynomial, so this polynomial must
be the zero polynomial. Therefore all the
a
i
are 0 and we conclude that the
ω
r
are
linearly independent.
(b) Note that
Φ
n
(
x
)
and
Φ
n
/
p
(
x
p
)
divide
x
n

1, hence the roots of these two poly
nomials are distinct. Let
α
,
β
∈
C
. Then
α
satisﬁes
Φ
n
(
x
)
if and only if it is
a primitive
n
th root of 1, and
α
satisﬁes
Φ
n
/
p
(
x
p
)
if and only if
β
p
is a primi
tive
n
/
p
th root of 1. Suppose
β
p
is a primitive
n
/
p
th root of 1. Then certainly
β
n
=
1, so
β
is an
n
th root of 1. Suppose
m
is a positive integer and
β
m
=
1.
Then
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 Fall '07
 PALinnell
 Math, Algebra, Polynomials, Complex number, Cyclic group, 5x3

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