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Multiplication in a finite field is multiplication modulo an irreducible reducing polynomial used to define the finite field. (I.e., it is multiplication followed by division using the reducing polynomial as the divisor—the remainder is the product.) The symbol "•" may be used to denote multiplication in a finite field.
A finite field is a finite set that 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. The number of elements of a finite field is called its order or, sometimes, its size.
Finite fields (also called Galois fields) are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory example F 4 is a field with four elements. Its subfield F 2 is the smallest field, because by definition a field has at least two distinct elements, 0 and 1.
In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a and its absolute Galois group. It is an interdisciplinary subject as it uses tools from algebraic number theory , arithmetic geometry , algebraic geometry , model theory , the theory of finite groups and of profinite groups .
For sufficiently small finite fields, a table of Zech logarithms allows an especially efficient implementation of all finite field arithmetic in terms of a small number of integer addition/subtractions and table look-ups. The utility of this method diminishes for large fields where one cannot efficiently store the table.
The elements of GF(2 n), i.e. a finite field whose order is a power of two, are usually represented as polynomials in GF(2)[X]. Multiplication of two such field elements consists of multiplication of the corresponding polynomials, followed by a reduction with respect to some irreducible polynomial which is taken from the construction of the ...
GF(2) (also denoted , Z/2Z or /) is the finite field with two elements. [1] [a]GF(2) is the field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity are denoted respectively 0 and 1, as usual.
Completing a number field K at a place w gives a complete field. If the valuation is Archimedean, one obtains R or C, if it is non-Archimedean and lies over a prime p of the rationals, one obtains a finite extension /: a complete, discrete valued field with finite residue field. This process simplifies the arithmetic of the field and allows the ...