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The polynomial P = x 4 + 1 is irreducible over Q but not over any finite field. On any field extension of F 2 , P = ( x + 1) 4 . On every other finite field, at least one of −1, 2 and −2 is a square, because the product of two non-squares is a square and so we have
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.
Given an n-bit message m 0,...,m n-1, we view it as a polynomial of degree n-1 over the finite field GF(2). = + + … +We then pick a random irreducible polynomial of degree k over GF(2), and we define the fingerprint of the message m to be the remainder () after division of () by () over GF(2) which can be viewed as a polynomial of degree k − 1 or as a k-bit number.
Definition. For a generic degree reducible monic polynomial equation of the form , where and are polynomials and does not vanish at , the Tschirnhaus transformation is the function: Such that the new equation in , , has certain special properties, most commonly such that some coefficients, , are identically zero. [2][3]
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 ...