Noun
Galois field (plural Galois fields)
(algebra) A finite field.
The Galois field
G
F
(
p
n
)
{\displaystyle \mathrm {GF} (p^{n})}
has order
p
n
{\displaystyle p^{n}}
and characteristic
p
{\displaystyle p}
.
The Galois field
G
F
(
p
n
)
{\displaystyle \mathrm {GF} (p^{n})}
is a finite extension of the Galois field
G
F
(
p
)
{\displaystyle \mathrm {GF} (p)}
and the degree of the extension is
n
{\displaystyle n}
.
The multiplicative subgroup of a Galois field is cyclic.
A Galois field
F
p
n
{\displaystyle \mathbb {F} _{p^{n}}}
is isomorphic to the quotient of the polynomial ring
F
p
{\displaystyle \mathbb {F} _{p}}
adjoin
x
{\displaystyle x}
over the ideal generated by a monic irreducible polynomial of degree
n
{\displaystyle n}
. Such an ideal is maximal and since a polynomial ring is commutative then the quotient ring must be a field. In symbols:
F
p
n
≅
F
p
[
x
]
(
f
^
n
(
x
)
)
{\displaystyle \mathbb {F} _{p^{n}}\cong {\mathbb {F} _{p}[x] \over ({\hat {f}}_{n}(x))}}
.