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College of the Holy Cross, Fall 2009
Math 351 – Homework 4 selected solutions
Chapter 16, #10.
To show that
R
[
x
] and
S
[
x
] are isomorphic, we need to deﬁne a mapping from
R
[
x
] to
S
[
x
] and show that it is a ring homomorphism, 1-1, and onto. What we know is that
R
and
S
are isomorphic, which means there exists a map
φ
:
R
→
S
that is a ring homom., 1-1, and onto.
Let’s deﬁne a map
δ
:
R
[
x
]
→
S
[
x
] that takes a polynomial
r
n
x
n
+
···
+
r
1
x
+
r
0
and maps it to
φ
(
r
n
)
x
n
+
···
+
φ
(
r
1
)
x
+
φ
(
r
0
). Note that this really gives a map
R
[
x
]
→
S
[
x
] since a typical element
of
R
[
x
] is a polynomial with coeﬃcients in
R
, and we’ve said how to map that to a polynomial with
coeﬃcients in
S
.
Since
R
[
x
] is not a quotient ring, we don’t need to worry about
δ
being well-deﬁned. Let’s show
that
δ
is a homomorphism. We have
δ
((
a
n
x
n
+
···
+
a
1
x
+
a
0
) + (
b
m
x
n
+
···
+
b
1
x
+
b
0
)) =
δ
(
c
s
x
s
+
···
+
c
1
x
+
c
0
)
,
where

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