Math 5125
Monday, November 14
Tenth Homework Solutions
1. (a) Both sides have degree
n
and both sides have roots
{
ζ
i

i
=
1
, . . . ,
n
}
. Since both
sides have constant term, it follows that both sides are equal.
(b) More generally in
C
[
x
,
y
]
, we have 1

(
x
/
y
)
n
=
∏
n
i
=
1
(
1

ζ
i
x
/
y
)
and we see that
y
n

x
n
=
∏
n
i
=
1
(
y

ζ
i
x
)
. Since
f
n
=
h
n

g
n
, we deduce that
f
n
=
∏
n
i
=
1
(
h

ζ
i
g
)
.
Suppose a prime
k
∈
C
[
x
]
divides two of the factors, say
(
h

ζ
i
g
)
,
(
h

ζ
j
g
)
where
1
≤
i
,
j
≤
n
,
i
6
=
j
(so
ζ
i
6
=
ζ
j
). Then
k
divides
(
ζ
i

ζ
j
)
g
and we deduce that
k
divides
g
,
h
, contrary to the hypothesis that
g
,
h
are relatively prime.
(c) This follows from the fact that
C
[
x
]
is a UFD. Indeed suppose
f
=
k
a
1
1
. . .
k
a
d
d
,
where the
k
i
are distinct primes and
a
i
≥
1. Then for each
i
,
k
i
divides precisely
one of the
(
h

ζ
j
)
and none of the others, by relative primeness. We deduce that
k
na
i
i
divides
(
h

ζ
j
)
and the result follows.
(d) By (c), we may write
h

g
=
a
n
,
h

ζ
g
=
b
n
,
h

ζ
2
g
=
c
n
, where
a
,
b
,
c
∈
C
[
x
]
(we use
n
≥
3 here). Since
h

g
,
h

ζ
g
,
h

ζ
2
g
are linearly dependent over