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Math 55  Homework 6
Due Friday, July 22, 2011
Directions:
•
You should work on these problems and write up solutions in
groups of two
.
•
Your answer should be clear enough that it explains to
someone who does not already
understand the answer
why it works. (This is diﬀerent than just convincing the grader
that you understand.)
1. Polynomials mod
p
.
In this problem:
•
p
is a prime.
•
all polynomials have integer coeﬃcients.
Deﬁnitions:
•
For polynomials
f
(
x
) =
a
k
x
k
+
...
+
a
1
x
+
a
0
and
g
(
x
) =
b
k
x
k
+
...
+
b
1
x
+
b
0
, we
say that
f
(
x
)
≡
g
(
x
)(mod
p
) if for every
i
,
a
i
≡
b
i
(mod
n
).
•
For a polynomial
f
(
x
) =
a
k
x
k
+
a
k

1
x
k

1
+
...
+
a
x
+
a
0
, deﬁne
deg
p
(
f
) to be the
largest integer (at most
k
) so that
p
6 
a
deg
p
(
f
)
(and
deg
p
(
f
) =
∞
if no such integer
exists.)
(a) If
f
(
x
)
≡
g
(
x
)(mod
p
), and
m
is an integer, prove that
f
(
m
)
≡
g
(
m
)(mod
p
).
(b) Prove the for any polynomial
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This note was uploaded on 03/14/2012 for the course MATH 55 taught by Professor Strain during the Summer '08 term at University of California, Berkeley.
 Summer '08
 STRAIN
 Math

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