Chapter
7
Introduction
to
ﬁnite
ﬁelds
This
chapter
provides
an
introduction
to
several
kinds
of
abstract
algebraic
structures,
partic-
ularly
groups,
ﬁelds,
and
polynomials.
Our
primary
interest
is
in
ﬁnite
ﬁelds,
i.e.,
ﬁelds
with
a
ﬁnite
number
of
elements
(also
called
Galois
ﬁelds).
In
the
next
chapter,
ﬁnite
ﬁelds
will
be
used
to
develop
Reed-Solomon
(RS)
codes,
the
most
useful
class
of
algebraic
codes.
Groups
and
polynomials
provide
the
requisite
background
to
understand
ﬁnite
ﬁelds.
A
ﬁeld
is
more
than
just
a
set
of
elements:
it
is
a
set
of
elements
under
two
operations,
called
addition
and
multiplication,
along
with
a
set
of
properties
governing
these
operations.
The
addition
and
multiplication
operations
also
imply
inverse
operations
called
subtraction
and
division.
The
reader
is
presumably
familiar
with
several
examples
of
ﬁelds,
such
as
the
real
ﬁeld
R
,
the
complex
ﬁeld
C
,
the
ﬁeld
of
rational
numbers
Q
,
and
the
binary
ﬁeld
F
2
.
7.1
Summary
In
this
section
we
brieﬂy
summarize
the
results
of
this
chapter.
The
main
body
of
the
chapter
will
be
devoted
to
deﬁning
and
explaining
these
concepts,
and
to
proofs
of
these
results.
Foreachpr
ime
p
and
positive
integer
m
≥
1,
there
exists
a
ﬁnite
ﬁeld
F
p
m
with
p
m
elements,
m
and
there
exists
no
ﬁnite
ﬁeld
with
q
elements
if
q
is
not
a
prime
power.
Any
two
ﬁelds
with
p
elements
are
isomorphic.
The
integers
modulo
p
form
a
prime
ﬁeld
F
p
under
mod-
p
addition
and
multiplication.
The
polynomials
F
p
[
x
]
over
F
p
modulo
an
irreducible
polynomial
g
(
x
)
∈
F
p
[
x
]o
fdeg
ree
m
form
a
ﬁnite
ﬁeld
with
p
m
elements
under
mod-
g
(
x
)
addition
and
multiplication.
For
every
prime
p
,
there
exists
at
least
one
irreducible
polynomial
g
(
x
)
∈
F
p
[
x
feachpos
it
ivedegree
m
≥
1,
so
all
ﬁnite
ﬁelds
may
be
constructed
in
this
way.
Under
addition,
F
p
m
is
isomorphic
to
the
vector
space
(
F
p
)
m
.
Under
multiplication,
the
nonzero
m
−
2
elements
of
F
p
m
form
a
cyclic
group
{
1
,α,.
..,α
p
}
generated
by
a
primitive
element
α
∈
F
p
m
.
The
elements
of
F
p
m
are
the
p
m
roots
of
the
polynomial
x
p
m
−
x
∈
F
p
[
x
].
The
polynomial
m
x
p
−
x
is
the
product
of
all
monic
irreducible
polynomials
g
(
x
)
∈
F
p
[
x
]
such
that
deg
g
(
x
)
divides
m
.
The
roots
of
a
monic
irreducible
polynomial
g
(
x
)
∈
F
p
[
x
]
form
a
cyclotomic
coset
of
deg
g
(
x
)e
lementso
f
F
p
m
which
is
closed
under
the
operation
of
raising
to
the
p
th
power.
every
n
that
divides
m
,
F
p
m
contains
a
subﬁeld
with
p
n
elements.
75