Finite field

From Example Problems
Jump to: navigation, search

In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory. The finite fields are completely known, as will be described below.

The complete list

Since every field of characteristic 0 contains the rationals and is therefore infinite, all finite fields have prime characteristic. However the converse is not true — there exist infinite fields of prime characteristic.

If p is a prime, the integers modulo p form a field with p elements, denoted Z/pZ, Fp or GF(p). (Sometimes Zp is used, but this can cause confusion with the ring of p-adic integers.) Every other field with p elements is isomorphic to this one.

If q = pn is a prime power, then there exists up to isomorphism exactly one field with q elements, written as Fq or GF(q). It can be constructed as follows: find an irreducible polynomial f(T) of degree n with coefficients in GF(p), then define GF(q) = GF(p)[T] / <f(T)>. Here, GF(p)[T] denotes the ring of all polynomials with coefficients in GF(p), <f(T)> denotes the ring ideal generated by f(T), and the quotient is meant in the sense of factor rings - the set of polynomials with coefficients in GF(p) on division by f(T). The polynomial f(T) can be found by factoring the polynomial T q-T over GF(p). The field GF(q) contains GF(p) as a subfield.

There are no other finite fields.

Examples

The polynomial f(T) = T 2 + T + 1 is irreducible over GF(2), and GF(4) = GF(2)[T] / <T2+T+1> can therefore be written as the set {0, 1, t, t+1} where the multiplication is defined (modularly) by t2 + t + 1 = 0. For example, to determine t3, note that t(t2 + t + 1) = 0; so t3 + t2 + t = 0, and thus t3 + t2 + t + 1 = 1, so t3 = 1. Similarly, since the characteristic of the field is 2 - coefficients are in GF(2), we can calculate powers of t in this instance by noting first that t2+t+1=0, and then t2=t+1. Then t3 = t(t2) = t(t+1) = t2+t = (t+1)+t = 1 as before.

In order to find the multiplicative inverse of t in this field, we have to find a polynomial p(T) such that T * p(T) = 1 modulo T 2 + T + 1. The polynomial p(T) = T + 1 works, and hence 1/t = t + 1. Note that the field GF(4) is completely unrelated to the ring Z4 of integers modulo 4.

To construct the field GF(27), we start with the irreducible polynomial T 3 + T 2 + T - 1 over GF(3). We then have GF(27) = {at2 + bt + c : a, b, c in GF(3)}, where the multiplication is defined by t 3 + t 2 + t - 1 = 0, or working from the rearrangement of the above in isolating the t3 term.

Properties and facts

If F is a finite field with q = pn elements (where p is prime), then

xq = x

for all x in F. Furthermore, the map

f : FF

defined by

f(x) = xp

is bijective and a homomorphism, and is therefore an automorphism. It is called the Frobenius automorphism, after Ferdinand Georg Frobenius.

The Frobenius automorphism has order n, so that the cyclic group it generates is the full group of automorphisms of the field.

The field GF(pm) contains a copy of GF(pn) if and only if n divides m. The reason for the if direction is that there exist irreducible polynomials of every degree over GF(pm).

If we actually construct our finite fields in such a fashion that GF(pn) is contained in GF(pm) whenever n divides m, then we can form the union of all these fields. This union is also a field, albeit an infinite one. It is the algebraic closure of each of the fields GF(pn). Even if we don't construct our fields this way, we can still speak of the algebraic closure, but some more delicacy is required in its construction.

Applications

The multiplicative group of every finite field is cyclic, a special case of a theorem mentioned in the article about fields. This means that if F is a finite field with q elements, then there always exists an element x in F such that F = { 0, 1, x, x2, ..., xq-2 }.

The element x is not unique. If we fix one, then for any non-zero element a in Fq, there is a unique integer n in {0, ..., q - 2} such that a = xn. The value of n for a given a is called the discrete log of a (in the given field, to base x). In practice, although calculating xn is relatively trivial given n, finding n for a given a is (under current theories) a computationally difficult process, and so has many applications in cryptography.

Finite fields also find applications in coding theory: many codes are constructed as subspaces of vector spaces over finite fields.

The first few finite fields

GF(2):

 + | 0 1        · | 0 1
 --+----        --+----
 0 | 0 1        0 | 0 0
 1 | 1 0        1 | 0 1
 

GF(3):

 + | 0 1 2       · | 0 1 2
 --+------       --+------
 0 | 0 1 2       0 | 0 0 0
 1 | 1 2 0       1 | 0 1 2
 2 | 2 0 1       2 | 0 2 1
 

GF(4):

 + | 0 1 A B       · | 0 1 A B
 --+--------       --+--------
 0 | 0 1 A B       0 | 0 0 0 0
 1 | 1 0 B A       1 | 0 1 A B
 A | A B 0 1       A | 0 A B 1
 B | B A 1 0       B | 0 B 1 A

See also

fr:Corps fini ja:有限体 nl:Eindig lichaam zh:有限域 he:שדה סופי ko:유한체