Math 5125
Monday, November 14
November 14, Ungraded Homework
Exercise 10.4.4 on page 375
Show that
Q
⊗
Z
Q
and
Q
⊗
Q
Q
are isomorphic as left
Q

modules.
Deﬁne a
Z
balanced map
θ
:
Q
×
Q
→
Q
⊗
Q
Q
by
θ
(
p
,
q
) =
p
⊗
q
. This induces a group
homomorphism
ˆ
θ
:
Q
⊗
Z
Q
→
Q
⊗
Q
Q
satisfying
ˆ
θ
(
p
⊗
q
) =
p
⊗
q
. It is routine to check
that this is actually a
Q
module map.
Next note that in
Q
⊗
Z
Q
, we have
px
⊗
q
=
p
⊗
xq
for all
x
∈
Q
; certainly this is true for
all
x
∈
Z
, but for the general rational number
x
, we need some more explanation. Write
x
=
a
/
b
, where
a
,
b
∈
Z
and
b
6
=
0. Then
px
⊗
q
=
p
(
a
/
b
)
⊗
q
=
p
/
b
⊗
aqb
/
b
=
p
⊗
q
(
a
/
b
) =
p
⊗
xq
.
It now follows that we can deﬁne a
Q
balanced map
φ
:
Q
×
Q
→
Q
⊗
Z
Q
by
φ
(
p
,
q
) =
p
⊗
q
(note that
φ
(
px
,
q
) =
φ
(
p
,
xq
)
for all
x
∈
Q
by the previous paragraph), so
φ
induces a
Z

module map
ˆ
φ
→
Q
⊗
Q
Q
→
Q
⊗
Z
Q
satisfying
ˆ
φ
(
p
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 Fall '07
 PALinnell
 Math, Algebra, Prime number, Cyclic group, Group isomorphism, Sylow, homomorphism θ

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