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In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF(q) is called a primitive element if it is a primitive (q − 1) th root of unity in GF(q); this means that each non-zero element of GF(q) can be written as α i for some natural number i.
The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group. In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.
Leonardo of Pisa (c. 1170 – c. 1250) described this method [1] [2] for generating primitive triples using the sequence of consecutive odd integers ,,,,, … and the fact that the sum of the first n terms of this sequence is .
Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. Here units refers to elements with a multiplicative inverse, which, in this ring, are exactly those coprime to n.
If the modulus and conductor are equal the character is primitive, otherwise imprimitive. An imprimitive character is induced by the character for the smallest modulus: χ 16 , 9 {\displaystyle \chi _{16,9}} is induced from χ 8 , 5 {\displaystyle \chi _{8,5}} and χ 16 , 15 {\displaystyle \chi _{16,15}} and χ 8 , 7 {\displaystyle \chi _{8,7 ...
Other Indo-European languages name man for his mortality, *mr̥tós meaning ' mortal ', so in Armenian mard, Persian mard, Sanskrit marta and Greek βροτός meaning ' mortal, human '. This is comparable to the Semitic word for ' man ' , represented by Arabic insan إنسان (cognate with Hebrew ʼenōš אֱנוֹשׁ ), from a root for ...
A generator for this cyclic group is a primitive n th root of unity. The n th roots of unity form an irreducible representation of any cyclic group of order n . The orthogonality relationship also follows from group-theoretic principles as described in Character group .
Although the Mersenne Twister pseudo-random number generator does not use a trinomial, it does take advantage of this. Richard Brent has been tabulating primitive trinomials of this form, such as x 74207281 + x 30684570 + 1. [5] [6] This can be used to create a pseudo-random number generator of the huge period 2 74207281 − 1 ≈ 3 × 10 22 ...