Galois field
finite field
A field with a finite number of elements. First considered by E. Galois [1].
The number of elements of any Galois field is a power of a prime number
, which is the characteristic of this field. For any prime number
and any natural number
there exists a (unique up to an isomorphism) field of
elements. It is denoted by
or by
. The field
contains the field
as a subfield if and only if
is divisible by
. In particular, any field
contains the field
, which is called the prime field of characteristic
. The field
is isomorphic to the field
of residue classes of the ring of integers modulo
. In any fixed algebraic closure
of
there exists exactly one subfield
for each
. The correspondence
is an isomorphism between the lattice of natural numbers with respect to division and the lattice of finite algebraic extensions (in
) of
with respect to inclusion. The lattice of finite algebraic extensions of any Galois field within its fixed algebraic closure is such a lattice.
The algebraic extension is simple, i.e. there exists a primitive element
such that
. Such an
will be any root of any irreducible polynomial of degree
from the ring
. The number of primitive elements of the extension
equals
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where is the Möbius function. The additive group of the field
is naturally endowed with the structure of an
-dimensional vector space over
. As a basis one may take
. The non-zero elements of
form a multiplicative group,
, of order
, i.e. each element of
is a root of the polynomial
. The group
is cyclic, and its generators are the primitive roots of unity of degree
, the number of which is
, where
is the Euler function. Each primitive root of unity of degree
is a primitive element of the extension
, but the converse is not true. More exactly, out of the
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irreducible unitary polynomials of degree over
there are
polynomials of which the roots are generators of
.
The set of elements of coincides with the set of roots of the polynomial
in
, i.e.
is characterized as the subfield of elements from
that are invariant with respect to the automorphism
, which is known as the Frobenius automorphism. If
, the extension
is normal (cf. Extension of a field), and its Galois group
is cyclic of order
. The automorphism
may be taken as the generator of
.
References
[1] | E. Galois, "Écrits et mémoires d'E. Galois" , Gauthier-Villars (1962) |
[2] | B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German) |
[3] | N.G. [N.G. Chebotarev] Tschebotaröw, "Grundzüge der Galois'schen Theorie" , Noordhoff (1950) (Translated from Russian) |
[4] | N. Bourbaki, "Algebra" , Elements of mathematics , 1 , Springer (1989) pp. Chapt. 1–3 (Translated from French) |
Galois field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Galois_field&oldid=30280