Eisenstein Criterion for Irreducibility of a
Polynomial
: Algebra
Jonathan L.F. King
University of Florida, Gainesville FL 326112082, USA
[email protected]
14 February, 2008
(at
09:47
)
Abstract:
Proofs of Eisenstein Criterion and the Gauss
Lemma.
Nomenclature.
Use
*
poly
for “polynomial” and
coeff
for “coefficient”.
I will be considering three
polys,
α
=
A
0
+
A
1
x
+
A
2
x
2
+
· · ·
+
A
J
x
J
,
β
=
B
0
+
B
1
x
+
· · ·
+
B
K
x
K
,
μ
=
M
0
+
M
1
x
+
M
2
x
2
+
· · ·
+
M
L
x
L
,
with each of
A
J
, B
K
, M
L
nonzero.
1
:
My convention is that later coefficients are all defined,
and equal zero; so 0 =
M
L
+1
=
M
L
+2
, . . .
. An
int
poly
is a poly whose coefficients are integers. A
rat
poly
has rational coefficients. Traditionally, the sym
bol
Z
[
x
] is used for the set of intpolys, and
Q
[
x
] for
the collection of ratpolys. These sets are rings.
A poly
α
is
“
a
unit
”
in
Z
[
x
], if its reciprocal is
also in
Z
[
x
].
There are only two units in
Z
[
x
]; the
constant polys
±
1. In
Q
[
x
], however, each constant
poly
x
→
q
is a unit, where
q
ranges over the nonzero
*
Use
≡
N
to mean “congruent mod
N
”.
Let
n
⊥
k
mean
that
n
and
k
are coprime.
Use
k
•
n
for “
k
divides
n
”.
Its
negation
k

n
means “
k
does not
divide
n
.”
Use
n
•
k
and
n

k
for “
n
is/isnot a multiple of
k
.” Finally, for
p
a prime
and
E
a natnum: Use doubleverticals,
p
E
•
n
, to mean that
E
is the
highest
power of
p
which divides
n
.
Or write
n
•
p
E
to emphasize that this is an assertion about
n
. Use
PoT
for
Power of Two
and
PoP
for
Power of (
a
) Prime
.
rationals.
A poly
μ
is
“
irreducible
over
Z
”
if, whenever it
can be factored into intpolys
μ
=
αβ
, then either
α
or
β
is a unit (
in
Z
[
x
]
). Thus 6
x

15 is reducible over
Z
,
since
6
x

15
=
3
·
[2
x

5]
.
2
:
However 6
x

15 is ir
reducible over
Q
, since 3 is a
Q
unit.
Reversing a poly.
Below is a “trivial” factoring re
sult; I put it here so that we can apply the Eisenstein
criterion to it later.
Say that a poly
μ
is
good
if its constant term is
nonzero. The
reversal
of a deg
L
good
μ
is
M
L
+
M
L

1
x
+
M
L

2
x
2
+
· · ·
+
M
1
x
L

1
+
M
0
x
L
,
denoted by
←

μ
. On the set of good polys, “reversal”
is an involution.
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 Fall '07
 JURY
 Calculus, Algebra, Prime number, Rational number, Eisenstein, gauss lemma, Prof. JLF King

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