Cryptography Reference
In-Depth Information
Now we turn to some facts about fields themselves. For the following exam-
ple, recall that a
finite field
is a field with a finite number of elements
n
∈
N
,
denoted by
F
n
. In general, if
K
is a finite field, then
K
=
F
p
m
for some prime
p
and
m
F
p
is called the
prime subfield
of
K
. In general, a prime subfield is a field having no proper subfields, so
∈
N
, also called
Galois fields
. The field
Q
is
Z
Z
F
p
is the prime
the prime subfield of any field of characteristic 0 and
/p
=
field of any field
K
=
F
p
m
. Also, we have the following result.
Theorem A.21
(
Multiplicative Subgroups of Fields)
If
F
is any field and
F
∗
is a finite subgroup of the multiplicative subgroup
of nonzero elements of
F
, then
F
∗
is cyclic.
A.3
In particular, if
F
=
F
p
n
is a
finite field, then
F
∗
is a finite cyclic group, and a generator of
F
∗
is called a
primitive element
of
F
.
, then there exists a field with
p
m
elements that is unique up to isomorphism (see [168, Corollary C.19, page 398],
for instance). Related to this is the following notion, which we will need, for
instance, in Chapter 11 when we discuss coding theory. For
m
It also follows that if
p
is prime and
m
∈
N
,a
primitive
m
th
root of unity
is a complex number
α
such that
α
m
= 1, but
α
j
∈
N
= 1 for all
natural numbers
j<m
. Primitive roots play a vital role in the proofs of results
involving field extensions of various types, especially finite. For instance, it may
be shown that
F
p
(
α
) where
α
is a primitive (
p
m
1)th root of unity.
(This is related to Galois theory; see [168, Appendix C, pages 393-401]for an
overview of this elegant theory.)
F
p
m
=
−
✦
Action on Rings
Definition A.36
(
Morphisms of Rings)
If
R
and
S
are two rings and
f
:
R
S
is a function such that
f
(
ab
)=
f
(
a
)
f
(
b
)
, and
f
(
a
+
b
)=
f
(
a
)+
f
(
b
)
for all
a,b
→
∈
R
, then
f
is called a
ring
homomorphism
. If, in addition,
f
:
R
S
is an injection as a map of sets, then
f
is called a
ring monomorphism
. If a ring homomorphism
f
is a surjection as
a map of sets, then
f
is called a
ring epimorphism
. If a ring homomorphism
f
is
a bijection as a map of sets, then
f
is called a
ring isomorphism
, and
R
is said
to be
isomorphic
to
S
, denoted by
R
=
S
. Lastly,
ker(
f
)=
→
{
s
∈
S
:
f
(
s
)=0
}
is called the
kernel of
f
. Also,
f
is injective if and only if
ker(
f
)=
{
0
}
.
There is a fundamental result that we will need in the text.
In order to
describe it, we need the following notion.
A.3
Recallthatamultiplicativeabeliangroupis
cyclic
wheneverthegroupgeneratedbysome
g ∈ G
coincides
with
G
. Note that any group of prime order is cyclic and the product of two
cyclic groups of relatively prime order is also a cyclic group. Also, if
S
is a nonempty subset
of a group
G
, then the intersection of all subgroups of
G
containing
S
is called the subgroup
generated
by
S
.
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