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In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors.This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm.
For univariate polynomials over the rationals (or more generally over a field of characteristic zero), Yun's algorithm exploits this to efficiently factorize the polynomial into square-free factors, that is, factors that are not a multiple of a square, performing a sequence of GCD computations starting with gcd(f(x), f '(x)). To factorize the ...
In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as Galois fields). The algorithm consists mainly of matrix reduction and polynomial GCD computations. It was invented by Elwyn Berlekamp in 1967.
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 ...
A primitive polynomial is a polynomial over a unique factorization domain, such that 1 is a greatest common divisor of its coefficients. Let F be a unique factorization domain. A non-constant irreducible polynomial over F is primitive. A primitive polynomial over F is irreducible over F if and only if it is irreducible over the field of ...
As a consequence of Gauss's lemma, this amounts to solving the problem also over the rationals. The algorithm starts by finding factorizations over suitable finite fields using Hensel's lemma to lift the solution from modulo a prime p to a convenient power of p. After this the right factors are found as a subset of these.
In computational number theory, Evdokimov's algorithm, named after Sergei Evdokimov, is an algorithm for factorization of polynomials over finite fields.It was the fastest algorithm known for this problem, from its publication in 1994 until 2020. [1]
In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over the field with elements. The method was discovered by Elwyn Berlekamp in 1970 [ 1 ] as an auxiliary to the algorithm for polynomial factorization over finite fields.