A126
A.1
Basic Algebra
A.1.1
Fields
In doing coding theory it is advantageous for our alphabet to have a certain
amount of mathematical structure. We are familiar at the bit level with boolean
addition (EXCLUSIVE OR) and multiplication (AND) within the set
{
0
,
1
}
:
+
0
1
0
0
1
1
1
0
×
0
1
0
0
0
1
0
1
We wish to give other alphabets, particularly ﬁnite ones, a workable arithmetic.
The objects we study (and of which the set
{
0
,
1
}
together with the above
operations is an example) are called ﬁelds. A ﬁeld is basically a set that possesses
an arithmetic having (most of) the properties that we expect — the ability to
add, multiply, subtract, and divide subject to the usual laws of commutativity,
associativity, and distributivity. The typical examples are the ﬁeld of rational
numbers (usually denoted
Q
), the ﬁeld of real numbers
R
, and the ﬁeld of
complex numbers
C
; however as just mentioned not all examples are so familiar.
The integers do
not
constitute a ﬁeld because in general it is not possible to
divide one integer by another and have the result still be an integer.
A
ﬁeld
is, by deﬁnition, a set
F
, say, equipped with two operations, + (addi
ﬁeld
tion) and
·
(multiplication), which satisfy the following seven usual arithmetic
axioms:
(1) (Closure) For each
a
and
b
in
F
, the sum
a
+
b
and the product
a
·
b
are welldeﬁned members of
F
.
(2) (Commutativity) For all
a
and
b
in
F
,
a
+
b
=
b
+
a
and
a
·
b
=
b
·
a
.
(3) (Associativity) For all
a
,
b
, and
c
in
F
, (
a
+
b
) +
c
=
a
+ (
b
+
c
)
and (
a
·
b
)
·
c
=
a
·
(
b
·
c
).
(4) (Distributivity) For all
a
,
b
, and
c
in
F
,
a
·
(
b
+
c
) =
a
·
b
+
a
·
c
and (
a
+
b
)
·
c
=
a
·
c
+
b
·
c
.
(5) (Existence of identity elements) There are distinct elements 0 and
1 of
F
such that, for all
a
in
F
,
a
+0 = 0+
a
=
a
and
a
·
1 = 1
·
a
=
a
.
(6) (Existence of additive inverses) For each
a
of
F
there is an ele
ment

a
of
F
such that
a
+ (

a
) = (

a
) +
a
= 0.
(7) (Existence of multiplicative inverses) For each
a
of
F
that does
not equal 0, there is an element
a

1
of
F
such that
a
·
(
a

1
) =
(
a

1
)
·
a
= 1.
It should be emphasized that these common arithmetic assumptions are the
only ones we make. In particular we make no ﬂat assumptions about operations