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In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules.
This requires the property that the field trace Tr L/K provides a non-degenerate quadratic form over K. This can be guaranteed if the extension is separable ; it is automatically true if K is a perfect field , and hence in the cases where K is finite, or of characteristic zero.
The theory of finite fields, whose origins can be traced back to the works of Gauss and Galois, has played a part in various branches of mathematics.Due to the applicability of the concept in other topics of mathematics and sciences like computer science there has been a resurgence of interest in finite fields and this is partly due to important applications in coding theory and cryptography.
Apart from the trivial case of a zero-dimensional space over any field, a vector space over a field F has a finite number of elements if and only if F is a finite field and the vector space has a finite dimension. Thus we have F q, the unique finite field (up to isomorphism) with q elements. Here q must be a power of a prime (q = p m with p prime).
This field is called a finite field or Galois field with four elements, and is denoted F 4 or GF(4). [8] The subset consisting of O and I (highlighted in red in the tables at the right) is also a field, known as the binary field F 2 or GF(2) .
A finite projective space defined over such a finite field has q + 1 points on a line, so the two concepts of order coincide. Such a finite projective space is denoted by PG(n, q), where PG stands for projective geometry, n is the geometric dimension of the geometry and q is the size (order) of the finite field used to construct the geometry.
A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension. [2] The property of an extension being Galois behaves well with respect to field composition and intersection. [3]
By making a modular multiplicative inverse table for the finite field and doing a lookup. By mapping to a composite field where inversion is simpler, and mapping back. By constructing a special integer (in case of a finite field of a prime order) or a special polynomial (in case of a finite field of a non-prime order) and dividing it by a. [7]