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 nonzero 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 ...

Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ...
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Primitive Polynomial (field Theory)
In field theory (mathematics), finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial (field theory), minimal polynomial of a primitive element (finite field), primitive element of the finite field . This means that a polynomial of degree with coefficients in is a ''primitive polynomial'' if it is monic polynomial, monic and has a root in such that \ is the entire field . This implies that is a primitive root of unity, primitive ()-root of unity in . Properties * Because all minimal polynomials are irreducible polynomial, 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 degre ...
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