Primitive Polynomial (other)
In different branches of mathematics, primitive polynomial may refer to: * Primitive polynomial (field theory), a minimal polynomial of an extension of finite fields * Primitive polynomial (ring theory) In algebra, the content of a polynomial with integer coefficients (or, more generally, with coefficients in a unique factorization domain) is the greatest common divisor of its coefficients. The primitive part of such a polynomial is the quotient ...

Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ...
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Primitive Polynomial (field Theory)
In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field . This means that a polynomial of degree with coefficients in is a ''primitive polynomial'' if it has a root in such that is the entire field . This implies that is a primitive ()-root of unity in . Properties * Because all minimal polynomials are irreducible, all primitive polynomials are also irreducible. * A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by ''x''. Over GF(2), is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by (it has 1 as a root). * An irreducible polynomial ''F''(''x'') of degree ''m'' over GF(''p''), where ''p'' is prime, is a primitive polynomial if the smallest positive integer ''n'' such that ''F''(''x'') divides is . * Over GF(''p''''m'') t ...
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