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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]
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 *.
Choose a field K and a finite group G. Cayley's theorem says that G is (up to isomorphism) a subgroup of the symmetric group S on the elements of G. Choose indeterminates {x α}, one for each element α of G, and adjoin them to K to get the field F = K({x α}). Contained within F is the field L of symmetric rational functions in the {x α}.
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.
In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF(p m).This means that a polynomial F(X) of degree m with coefficients in GF(p) = Z/pZ is a primitive polynomial if it is monic and has a root α in GF(p m) such that {,,,,, …} is the entire field GF(p m).
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).
In mathematics, local class field theory, introduced by Helmut Hasse, [1] is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of ...