MATH 4107
PRACTICE PROBLEMS WITH SOLUTIONS
April 15, 2007
Here are a few practice problems on rings. Try to work these WITHOUT looking at the
solutions! After you write your own solution, you can compare to my solution. Your solution
does not need to be identical—there are often many ways to solve a problem—but it does
need to be CORRECT.
1. Work out the rule of computation in the ring
R
[
x
]
/
(
f
), where
f
(
x
) =
x
3

1. Note that
the quotient ring consists of elements
a
+
bx
+
cx
2
+ (
f
).
Solution
For easier notation, let
I
= (
f
). Addition is easy:
(
a
1
+
b
1
x
+
c
1
x
2
+
I
)
+
(
a
2
+
b
2
x
+
c
2
x
2
+
I
)
=
(
a
1
+
a
2
) + (
b
1
+
b
2
)
x
+ (
c
1
+
c
2
)
x
2
+
I.
Multiplication is harder.
The key fact is that
x
3

1 is in
I
, so
x
3

1 +
I
=
I
.
Therefore
x
3
+
I
= 1 +
I
. Similarly,
x
4
+
I
=
xx
3
+
I
=
(
x
+
I
)(
x
3
+
I
)
=
(
x
+
I
)(
1 +
I
)
=
x
+
I.
Hence,
(
a
1
+
b
1
x
+
c
1
x
2
+
I
)(
a
2
+
b
2
x
+
c
2
x
2
+
I
)
=
a
1
a
2
+ (
b
1
a
2
+
a
1
b
2
)
x
+ (
c
1
a
2
+
b
1
b
2
+
a
1
c
2
)
x
2
+ (
a
1
c
2
+
c
1
a
2
)
x
3
+
c
1
c
2
x
4
+
I
=
a
1
a
2
+ (
b
1
a
2
+
a
1
b
2
)
x
+ (
c
1
a
2
+
b
1
b
2
+
a
1
c
2
)
x
2
+ (
a
1
c
2
+
c
1
a
2
) +
c
1
c
2
x
+
I
=
(
a
1
a
2
+
a
1
c
2
+
c
1
a
2
) + (
b
1
a
2
+
a
1
b
2
+
c
1
c
2
)
x
+ (
c
1
a
2
+
b
1
b
2
+
a
1
c
2
)
x
2
+
I.
±
2.
Let
F
be a ±eld.
Show that a cubic polynomial in
F
[
x
] either has a root in
F
or is
irreducible over
F
.
Solution
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 Spring '10
 Staff
 Math, Algebra, Polynomials, Complex number, Degree of a polynomial, zp

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