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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentA.1. BASIC ALGEBRA
A127
called “subtraction” or “division”. These operations are best thought of as the
“undoing” respectively of addition and multiplication and, when desired, can be
deﬁned using the known operations and their properties. Thus subtraction can
be deﬁned by
a

b
=
a
+(

b
) using (6), and division deﬁned by
a/b
=
a
·
(
b

1
)
using (7) (provided
b
is not 0).
Other familiar arithmetic properties that are not assumed as axioms either
must be proven from the assumptions or may be false in certain ﬁelds. For
instance, it is not assumed but can be proven that always in a ﬁeld (

1)
·
a
=

a
.
(Try it!) A related, familiar result which can be proven for all ﬁelds
F
is
that, given
a
and
b
in
F
, there is always a unique solution
x
in
F
to the
equation
a
+
x
=
b
. On the other hand the properties of positive and/or negative
numbers familiar from working in the rational ﬁeld
Q
and the real ﬁeld
R
do
not have a place in the general theory of ﬁelds. Indeed there is no concept at
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 NA
 Linear Algebra, Vector Space, aZn

Click to edit the document details