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PEN Q Problems
8
Q 8
Q 8
Source:
May 25, 2007
algebra
polynomial
analytic geometry
graphing lines
slope
combinatorial geometry
Polynomials
Problem Statement
Show that a polynomial of odd degree
2
m
+
1
2m+1
2
m
+
1
over
Z
\mathbb{Z}
Z
,
f
(
x
)
=
c
2
m
+
1
x
2
m
+
1
+
⋯
+
c
1
x
+
c
0
,
f(x)=c_{2m+1}x^{2m+1}+\cdots+c_{1}x+c_{0},
f
(
x
)
=
c
2
m
+
1
x
2
m
+
1
+
⋯
+
c
1
x
+
c
0
,
is irreducible if there exists a prime
p
p
p
such that
p
∤
c
2
m
+
1
,
p
∣
c
m
+
1
,
c
m
+
2
,
⋯
,
c
2
m
,
p
2
∣
c
0
,
c
1
,
⋯
,
c
m
,
and
p
3
∤
c
0
.
p \not\vert c_{2m+1}, p \vert c_{m+1}, c_{m+2}, \cdots, c_{2m}, p^{2}\vert c_{0}, c_{1}, \cdots, c_{m}, \; \text{and}\; p^{3}\not\vert c_{0}.
p
∣
c
2
m
+
1
,
p
∣
c
m
+
1
,
c
m
+
2
,
⋯
,
c
2
m
,
p
2
∣
c
0
,
c
1
,
⋯
,
c
m
,
and
p
3
∣
c
0
.
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