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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 ...
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.
(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.
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.
Theorem statement. The primitive element theorem states: Every separable field extension of finite degree is simple. This theorem applies to algebraic number fields, i.e. finite extensions of the rational numbers Q, since Q has characteristic 0 and therefore every finite extension over Q is separable. Using the fundamental theorem of Galois ...
To state it precisely, let F = GF(q) be the finite field with q elements, for some fixed q, and let N n be the number of monic irreducible polynomials over F whose degree is equal to n. That is, we are looking at polynomials with coefficients chosen from F, which cannot be written as products of polynomials of smaller degree. In this setting ...