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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.
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]
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.
The Fano plane, the projective plane over the field with two elements, is one of the simplest objects in Galois geometry.. Galois geometry (named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or Galois field). [1]
While there is a unique finite field of order p n up to isomorphism, the representation of the field elements depends on the choice of irreducible polynomial. The Conway polynomial is a way of standardizing this choice. The non-zero elements of a finite field F form a cyclic group under multiplication, denoted F *.
Let F q = GF(q) be the finite field of characteristic p, that is, the field having q elements where q = p e for some prime p.A polynomial f with coefficients in F q (symbolically written as f ∈ F q [x]) is a permutation polynomial of F q if the function from F q to itself defined by () is a permutation of F q.
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).