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
ReedSolomon
(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.
For each prime
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
] of degree
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
] of each positive degree
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
) elements of
F
p
m
which
is
closed
under
the
operation
of
raising
to
the
p
th
power.
For
every
n
that
divides
m
,
F
p
m
contains
a
subfield
with
p
n
elements.
75
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76
CHAPTER
7.
INTRODUCTION
TO
FINITE
FIELDS
For
further
reading
on
this
beautiful
subject,
see
[E.
R.
Berlekamp,
Algebraic
Coding
The
ory
,
Aegean
Press,
1984],
[R.
Lidl
and
H.
Niederreiter,
Introduction
to
Finite
Fields
and
their
Applications
,
Cambridge
University
Press,
1986]
or
[R.
J.
McEliece,
Finite
Fields
for
Com
puter
Scientists
and
Engineers
,
Kluwer,
1987],
[M.
R.
Schroeder,
Number
Theory
in
Science
and
Communication
,
Springer,
1986],
or
indeed
any
book
on
finite
fields
or
algebraic
coding
theory.
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 Winter '11
 PhanThuongCang
 Prime number, Cyclic group, finite field, fQ

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