Zech's logarithms are useful for large computations, such as linear algebra over medium-sized fields, that is, fields that are sufficiently large for making natural algorithms inefficient, but not too large, as one has to pre-compute a table of the same size as the order of the field.Įvery nonzero element of a finite field is a root of unity, as x q−1 = 1 for every nonzero element of GF( q). , q − 2 (it is convenient to define the discrete logarithm of zero as being −∞). In summary:Īllows one to solve this problem by constructing the table of the discrete logarithms of a n + 1, called Zech's logarithms, for n = 0. The structure theorem of finite abelian groups implies that this multiplicative group is cyclic, that is, all non-zero elements are powers of a single element. As the equation x k = 1 has at most k solutions in any field, q – 1 is the highest possible value for k. If the characteristic of a field is p, then the elements which can be written as sums of 1s form a p inside the field, i.e., a subfield. By Lagrange's theorem, there exists a divisor k of q – 1 such that x k = 1 for every non-zero x in GF( q). The set of non-zero elements in GF( q) is an abelian group under the multiplication, of order q – 1. One may therefore identify all finite fields with the same order, and they are unambiguously denoted F q with m less than and coprime with 15 (that is, 1, 2, 4, 7, 8, 11, 13, 14). Moreover, a field cannot contain two different finite subfields with the same order. The length property represents an unsigned, 32-bit integer that is always numerically greater than the highest index in the array. If q = p k, all fields of order q are isomorphic (see § Existence and uniqueness below). Any ideas why this is Because that's how the language works.

In a field of order p k, adding p copies of any element always results in zero that is, the characteristic of the field is p.

A finite field of order q exists if and only if q is a prime power p k (where p is a prime number and k is a positive integer). The number of elements of a finite field is called its order or, sometimes, its size.

6.2 Number of monic irreducible polynomials of a given degree over a finite fieldĪ finite field is a finite set which is a field this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the field axioms.

6.1 Irreducible polynomials of a given degree.

5 Frobenius automorphism and Galois theory.

Write number as element of field