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For b ∈ L, let F b be the map / (). Then F b ≠ F c if b ≠ c. Moreover, the K-linear transformations from L to K are exactly the maps of the form F b as b varies over the field L. When K is the prime subfield of L, the trace is called the absolute trace and otherwise it is a relative trace. [4]
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.
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The field L is then a finite-dimensional vector space over K. Multiplication by α, an element of L, : =, is a K-linear transformation of this vector space into itself. The norm, N L/K (α), is defined as the determinant of this linear transformation. [1]
The splitting field of x q − x over F p is the unique finite field F q for q = p n. [2] Sometimes this field is denoted by GF(q). The splitting field of x 2 + 1 over F 7 is F 49; the polynomial has no roots in F 7, i.e., −1 is not a square there, because 7 is not congruent to 1 modulo 4. [3]
This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups. Furthermore, if p {\displaystyle p} is the smallest prime number dividing the order of a finite group, G {\displaystyle G} , then if G / H {\displaystyle G\,/\,H} has order p {\displaystyle p} , H ...
The finite field with p n elements is denoted GF(p n) and is also called the Galois field of order p n, in honor of the founder of finite field theory, Évariste Galois. GF( p ), where p is a prime number, is simply the ring of integers modulo p .
Their first calculator with a microprocessor is the Busicom 141-PF. Their entry based calculators, the Busicom LE-120A (Handy-LE) and LE-120S (Handy) , [ 6 ] were the first to fit in a pocket and also the first calculators to use an LED display.