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Irreducible polynomials over finite fields are also useful for pseudorandom number generators using feedback shift registers and discrete logarithm over F 2 n. The number of irreducible monic polynomials of degree n over F q is the number of aperiodic necklaces, given by Moreau's necklace-counting function M q (n). The closely related necklace ...
Over GF(2), x + 1 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 x + 1 (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 ...
The Cantor–Zassenhaus algorithm takes as input a square-free polynomial (i.e. one with no repeated factors) of degree n with coefficients in a finite field whose irreducible polynomial factors are all of equal degree (algorithms exist for efficiently factoring arbitrary polynomials into a product of polynomials satisfying these conditions, for instance, () / ((), ′ ()) is a squarefree ...
For univariate polynomials over a field, this results from Bézout's identity, which itself results from the Euclidean algorithm. So, let R be a unique factorization domain, which is not a field, and R[X] the univariate polynomial ring over R. An irreducible element r in R[X] is either an irreducible element in R or an irreducible primitive ...
It follows that they are roots of irreducible polynomials of degree 6 over GF(2). This implies that, over GF(2), there are exactly 9 = 54 / 6 irreducible monic polynomials of degree 6. This may be verified by factoring X 64 − X over GF(2). The elements of GF(64) are primitive n th roots of unity for some n dividing 63.
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 field.
(A polynomial with integer coefficients is primitive if it has 1 as a greatest common divisor of its coefficients. [note 2]) A corollary of Gauss's lemma, sometimes also called Gauss's lemma, is that a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers. More generally, a primitive ...
Irreducibility (mathematics) In mathematics, the concept of irreducibility is used in several ways. A polynomial over a field may be an irreducible polynomial if it cannot be factored over that field. In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial.