In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of computer algebra systems.The first polynomial factorization algorithm was published by Theodor von Schubert in 1793. Leopold Kronecker rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension. But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems:When the long-known finite step algorithms were first put on computers, they turned out to be highly inefficient. The fact that almost any uni- or multivariate polynomial of degree up to 100 and with coefficients of a moderate size (up to 100 bits) can be factored by modern algorithms in a few minutes of co
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In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and po
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the op
In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This
An algorithm is presented to factorize polynomials in several variables with integral coefficients that is polynomial-time in the degrees of the polynomial to be factored, for any fixed number of variables. The algorithm generalizes the algorithm presented by A. K. Lenstra et al. to factorize integral polynomials in one variable.
In 1982 a polynomial-time algorithm for factoring polynomials in one variable with rational coefficients was published (see A.K. Lenstra, H.W. Lenstra, Jr. and L. Lovasz, Math. Ann., vol.261, p.515-34, 1982). This L3-algorithm came as a rather big surprise: hardly anybody expected that the problem allowed solution in polynomial time. The purpose of this note is to present an informal description of the L3-algorithm