27.
Fields from quotient rings
In Lecture 26 we have shown that the quotient ring
R
[
x
]
/
(
x
2
+ 1)
R
[
x
] is
isomorphic to
C
, so, in particular, it is a field, while
R
[
x
]
/
(
x
2

1)
R
[
x
] is
not a field.
The reason we did not get a field in the second case is clear:
the polynomial
x
2

1 is reducible, that is, has a nontrivial factorization
x
2

1 = (
x

1)(
x
+1), and we have seen in the proof from Example 3 that the
existence of factorization (
x

1)(
x
+ 1) is what prevents
R
[
x
]
/
(
x
2

1)
R
[
x
]
from being a field.
On the other hand,
x
2
+ 1 is irreducible, although it
is not clear how to deduce that
R
[
x
]
/
(
x
2
+ 1)
R
[
x
] is a field just from the
irreducibility of
x
2
+ 1.
In this lecture we will settle the latter issue: we will show that if
F
is
any field and
p
∈
F
[
x
] is a polynomial, the the quotient ring
F
[
x
]
/pF
[
x
] is
a field
⇐⇒
p
is irreducible.
27.1.
Basic definitions.
Definition.
Let
F
be a field and
p
∈
F
[
x
] a polynomial with coefficients in
F
. Then
p
is called irreducible
if
(i)
p
is nonconstant, or, equivalently, deg(
p
)
>
0;
(ii)
p
does not have nontrivial factorizations, that is,
p
cannot be written
as
p
=
gh
where
g, h
∈
F
[
x
] and both
g
and
h
are nonconstant.
Remark:
Irreducible polynomials are direct counterparts of prime inte
gers.
The convention not to consider constant polynomials as irreducible
corresponds to the convention not to consider 1 as a prime number. As we
will see shortly, the analogy between prime integers and irreducible polyno
mials goes well beyond the definition.
Definition.
Let
F
be a field and
p
∈
F
[
x
] a polynomial with coefficients in
F
. Then
p
is called monic
if the leading coefficient of
p
is equal to 1.
Next we define the greatest common divisor for polynomials.
Definition.
Let
F
be a field and
f, g
∈
F
[
x
] two polynomials. A polynomial
d
∈
F
[
x
] is called the greatest common divisor (gcd) of
f
and
g
if the
following conditions hold:
(i)
d
is monic
(ii)
d
divides both
f
and
g
1
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(iii) If
h
∈
F
[
x
] is another polynomial which divides both
f
and
g
, then
h
divides
d
.
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 Winter '11
 PhanThuongCang
 Prime number, finite field, Commutative ring, R/I

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