Graphics Reference
In-Depth Information
B.8
Fields
Definition.
Let R be a ring with unity. If every nonzero element of R is a unit, then
R is called a
skew field
or
division ring
. A
field
is a commutative division ring. A subring
of a field that is a field is called a
subfield
.
B.8.1. Example.
Q
,
R
, and
Z
p
, where p is prime, are all fields.
It is easy to see that the intersection of fields is again a field.
Definition.
Let K be a field. The intersection k of all of its nonzero subfields is called
the
prime field of K
. If k = K, then K is called a
prime field
.
The prime field of a field can also be described as the “smallest” subfield of a
field.
B.8.2. Theorem.
A prime field is isomorphic to either
Q
or
Z
p
, where p is
prime.
Proof.
See [Mill58].
Definition.
Let K be a field and k its prime subfield. If k is isomorphic to
Z
p
, where
p is prime, then K is said to have
characteristic
p. Otherwise, k is isomorphic to
Q
and
K is said to have
characteristic
0.
Every integral domain D can be imbedded in a field. The construction generalizes
the way that one gets the rational numbers from the integers. Let
=
(
{
)
0
Q
*
a b
,
a b
,
Œ
D and b
π
.
Define an equivalence relation ~ on Q* as follows:
(
)
(
)
a b
,
~,
c d if and only if ad
=
cd
.
It is easy to check that ~ is an equivalence relation. Let Q denote the equivalence
classes of Q* with respect to ~. Define two operations + and · on Q:
[
]
+
[
]
=+
[
]
a b
,
c d
,
ad
bc bd
,
[
]
◊
[
]
=
[
]
a b
,
c d
,
ac bd
,
.
B.8.3. Theorem.
(Q,+, ·) is a well-defined field.
Proof.
See [Mill58].
Definition.
The field (Q,+, ·) in Theorem B.8.3 is called the
quotient field
of the inte-
gral domain D.