A Primer on Finite Fields
In
(1)

(7)
F
will denote a finite field.
(1)
F
contains a copy of
Z
p
=
F
p
, for some prime
p
. (This prime is called
the
characteristic
of
F
.)
(2)
There is a positive integer
d
with

F

=
p
d
.
Proof.
From the definitions,
F
is a vector space over
F
p
. Let
e
1
, . . . ,
e
d
be a basis. Then
F
=
n
∑
d
i
=1
a
i
e
i
a
1
, . . . , a
d
∈
F
p
o
. Thus

F

is the number of choices for the
a
i
, namely
p
d
.
(3)
Let
α
∈
F
≥
F
p
. Then
F
p
[
α
] =
(
k
X
i
=0
a
i
α
i
k
≥
0
, a
i
∈
F
p
)
is a subring of
F
.
(That is, it is closed under addition, subtraction, and
multiplication.
See
(6)
and
(8)
, where we show that
F
p
[
α
] is actually a
subfield.)
(4)
Let
m
(
x
)
∈
F
p
[
x
] be a monic polynomial of minimal degree with
m
(
α
) =
0. (It exists since
F
is finite.) Then
F
p
[
α
] is a copy of
F
p
[
x
] (mod
m
(
x
)), that
is, the polynomial ring
F
p
[
x
] with arithmetic done modulo the polynomial
m
(
x
).
The polynomial
m
(
x
) is called the
minimal polynomial
of
α
over
F
p
and is uniquely
determined. We sometimes write
m
α
(
x
) or even
m
α,
F
p
(
x
) for the minimal polynomial of
α
over
F
p
.
Let
f
(
x
) be a nonconstant polynomial of
K
[
x
].
Then
f
(
x
) is called
irreducible
in
K
[
x
] if every factorization
f
(
x
) =
a
(
x
)
b
(
x
) in
K
[
x
] has
{
deg
a,
deg
b
}
=
{
0
,
deg
f
}
. (This
corresponds to prime numbers in
Z
.) Otherwise
f
(
x
) is
reducible
.
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 Spring '11
 NA
 Polynomials, Prime number, Complex number

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