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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]
It is a modification of finite element method. The method divides the domain concerned into sections of infinite length. In contrast with a finite element which is approximated by polynomial expressions on a finite support, the unbounded length of the infinite element is fitted with functions allowing the evaluation of the field at the asymptote.
finite extensions of Q p (local fields of characteristic zero) finite extensions of F p ((t)), the field of Laurent series over F p (local fields of characteristic p). These two types of local fields share some fundamental similarities. In this relation, the elements p ∈ Q p and t ∈ F p ((t)) (referred to as uniformizer) correspond to each ...
The use of base fields other than Q is crucial in many areas of mathematics. For example, in algebraic number theory, one often does Galois theory using number fields, finite fields or local fields as the base field. It allows one to more easily study infinite extensions.
There exist infinite fields of prime characteristic. For example, the field of all rational functions over /, the algebraic closure of / or the field of formal Laurent series / (()). The size of any finite ring of prime characteristic p is a power of p.
GF(2) is the only field with this property (Proof: if x 2 = x, then either x = 0 or x ≠ 0. In the latter case, x must have a multiplicative inverse, in which case dividing both sides by x gives x = 1. All larger fields contain elements other than 0 and 1, and those elements cannot satisfy this property).
For a field k, a finite-dimensional, unital, associative algebra is Frobenius if and only if the injective right A-module Hom k (A,k) is isomorphic to the right regular representation of A. For an infinite field k, a finite-dimensional, unital, associative k-algebra is a Frobenius algebra if it has only finitely many minimal right ideals.