Homework 10 Solutions
1. BWOC, suppose the highest degree irreducible polynomial in
F
[
x
] is
n
.
Since
F
is finite,
there are only a finite number of irreducible polynomials of each degree, and we don’t have
irreducible polynomials above degree
n
, thus our assumption implies that there are only a
finite number of irreducible polynomials in
F
[
x
]. Enumerate these as
p
1
(
x
)
, . . . p
m
(
x
) (and
m
≥
2 clearly since
x
and
x
+ 1 are irreducible in
F
[
x
]). Consider the polynomial
f
(
x
) =
p
1
(
x
)
p
2
(
x
)
· · ·
p
m
(
x
) + 1.
deg
(
f
(
x
))
> n
so
f
(
x
) is reducible.
Since
p
1
, . . . p
m
are the only
irreducible polynomials, one of these must divide
f
(
x
).
p
i

f
(
x
) and
p
i
(
x
)

p
1
(
x
)
p
2
(
x
)
· · ·
p
m
(
x
)
implies
p
i
(
x
)

1, a contradiction.
Thus there must be irreducible polynomials of arbitrarily
high degree.
2.
(a) 5
x
+ 1 = (2
x
+ 1)(3
x
+ 1)
(b) 5
x
= (4
x
+ 3)(3
x
+ 2)
(c) 2
x
+ 2 = (2
x
+ 2)(3
x
+ 1)
(d) 2
x
+ 2 = (2
x
+ 2)(3
x
2
+ 1)
3.
(a)
x
5
+ 9
x
4
+ 12
x
2
+ 6  Eisenstein’s Criterion with
p
= 3 works.
(b)
x
3

3
x
+ 3  Eisenstein’s Criterion with
p
= 3 works again.
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 Spring '11
 Johnson
 Algebra, Polynomials, Complex number, finite field, irreducible polynomials

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