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Math 523
Homework 4
Due 7/21/2008
Total: 40 points
From Section 2 of the lecture notes:
In these exercises, let
F
be a ﬁeld, let
p
(
x
) be a nonconstant polynomial in
F
[
x
], and let
R
=
F
[
x
]
/
h
p
(
x
)
i
.
1.
(5 pts) If
f
(
x
) is any polynomial, show that
h
[
f
(
x
)]
i
=
h
[
d
(
x
)]
i
in
R
, where
d
(
x
) = gcd(
f
(
x
)
, p
(
x
)).
2.
(5 pts) Suppose that
p
(
x
) =
g
(
x
)
h
(
x
), where gcd(
g
(
x
)
, h
(
x
)) = 1. Show that in
R
we have
h
[
g
(
x
)]
i
=
{
[
f
(
x
)]
∈
R

f
(
x
)
h
(
x
)
≡
0 (mod
p
(
x
))
}
.
4.
(5 pts) In
Z
2
[
x
]
/
±
x
15

1
²
, ﬁnd the idempotent generator for the ideal
±
x
4
+
x
+ 1
²
. Be sure to show that your
answer is in fact idempotent.
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Unformatted text preview: From Section 3 of the lecture notes: 1. (15 pts) Find the splitting ﬁelds over Z 2 for the following polynomials: (a) x 2 + x + 1 (b) x 2 + 1 (c) x 3 + x + 1 (d) x 3 + x 2 + 1 2. (5 pts) Find the splitting ﬁeld for x px over Z p . 3. (5 pts) Show that if F is an extension ﬁeld of K of degree 2, then F is the splitting ﬁeld over K for some polynomial....
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This note was uploaded on 06/13/2011 for the course CRYPTO 101 taught by Professor Na during the Spring '11 term at Harding.
 Spring '11
 NA

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