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.
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