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| In [[mathematics]] and [[computer algebra]], '''factorization of polynomials''' or '''polynomial factorization''' refers to factoring a [[polynomial]] with coefficients in a given [[field (mathematics)|field]] or in the [[integers]] into [[irreducible polynomial|irreducible factors]] with coefficients in same domain. Polynomial factorization is one of the fundamental tools of the [[computer algebra system]]s.
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| The history of polynomial factorization starts with [[Hermann Schubert]] who in 1793 described the first polynomial factorization algorithm, and [[Leopold Kronecker]], who 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]]. In a survey of the subject, Erich Kaltofen wrote in 1982 (see the bibliography, below):
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| <blockquote>
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| 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 computer time indicates how successfully this problem has been attacked during the past fifteen years.
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| </blockquote>
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| <p>
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| ==Formulation of the question==
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| [[Polynomial ring]]s over the integers or over a field are [[unique factorization domain]]s. This means that every element of these rings is a product of a constant and a product of [[irreducible polynomial]]s (those that are not the product of two non-constant polynomials). Moreover, this decomposition is unique up to multiplication of the factors by invertible constants.
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| Factorization depends on the base field. For example, the [[fundamental theorem of algebra]], which states that every polynomial with [[complex number|complex]] coefficients has complex roots, implies that a polynomial with integer coefficients can be factored (with [[root-finding algorithms]]) into [[linear function|linear factor]]s over the complex field '''C'''. Similarly, over the [[field of reals]], the irreducible factors have degree at most two, while there are polynomials of any degree that are irreducible over the [[field of rationals]] '''Q'''.
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| The question of polynomial factorization makes sense only for coefficients in a ''computable field'' whose every element may be represented in a computer and for which there are algorithms for the arithmetic operations. Fröhlich and Shepherson have provided examples of such fields for which no factorization algorithm can exist.
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| The fields of coefficients for which factorization algorithms are known include [[prime field]]s (i.e. the [[field of rationals]] and prime [[modular arithmetic]]) and their [[finitely generated field extension]]s. Integer coefficients are also tractable: Kronecker's method is interesting only from a historical point of view, modern algorithms proceed by a succession of:
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| * Square-free factorization
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| * Factorization over finite fields
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| and reductions:
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| * From the [[multivariate polynomial|multivariate]] case to the [[univariate]] one
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| * From coefficients in a [[purely transcendental extension]] to the multivariate case over the ground field (see [[#Primitive part–content factorization|below]])
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| * From coefficients in an algebraic extension to coefficients in the ground field
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| * From rational coefficients to integer coefficients (see [[#Primitive part–content factorization|below]])
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| * From integer coefficients to coefficients in a prime field with ''p'' elements, for a well chosen ''p''.
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| ==Primitive part–content factorization==<!-- There are two links here from a preceding section-->
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| {{See also|Content (algebra)|Gauss's lemma (polynomial)}}
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| In this section, we show that factoring over '''Q''' (the rational numbers) and over '''Z''' (the integers) is essentially the same problem.
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| The ''content'' of a polynomial ''p'' ∈ '''Z'''[''X''], denoted "cont(''p'')", is, [[up to]] its sign, the [[greatest common divisor]] of its coefficients. The ''primitive part'' of ''p'' is primpart(''p'')=''p''/cont(''p''), which is a [[primitive polynomial (ring theory)|primitive polynomial]] with integer coefficients. This defines a factorization of ''p'' into the product of an integer and a primitive polynomial. This factorization is unique up to the sign of the content. It is a usual convention to choose the sign of the content such that the leading coefficient of the primitive part is positive.
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| For example,
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| :<math>
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| -10x^2 + 5x + 5 = (-5)\cdot (2x^2 - x - 1) \,
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| </math>
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| is a factorization into content and primitive part.
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| Every polynomial ''q'' with rational coefficients may be written
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| :<math>q = \frac{p}{c},</math>
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| where ''p'' ∈ '''Z'''[''X''] and ''c'' ∈ '''Z''': it suffices to take for ''c'' a multiple of all denominators of the coefficients of ''q'' (for example their product) and ''p'' = ''cq''. The ''content'' of ''q'' is defined as:
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| :<math>\text{cont} (q) =\frac{\text{cont} (p)}{c},</math>
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| and the ''primitive part'' of ''q'' is that of ''p''. As for the polynomials with integer coefficients, this defines a factorization into a rational number and a primitive polynomial with integer coefficients. This factorization is also unique up to the choice of a sign.
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| For example,
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| :<math>
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| \frac{1}{3}x^5 + \frac{7}{2} x^2 + 2x + 1 = \frac{1}{6} ( 2x^5 + 21x^2 + 12x + 6)
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| </math>
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| is a factorization into content and primitive part.
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| [[Carl Friedrich Gauss|Gauss]] has first proved that the product of two primitive polynomials is also primitive ([[Gauss's lemma (polynomial)|Gauss's lemma]]). This implies that a primitive polynomial is irreducible over the rationals if and only if it is irreducible over the integers. This implies also that the factorization over the rationals of a polynomial with rational coefficients is the same as the factorization over the integers of its primitive part. On the other hand, the factorization over the integers of a polynomial with integer coefficients is the product of the factorization of its primitive part by the factorization of its content.
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| In other words, integer GDD computation allows to reduce the factorization of a polynomial over the rationals to the factorization of a primitive polynomial with integer coefficients, and to reduce the factorization over the integers to the factorization of an integer and a primitive polynomial.
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| Everything that precedes remains true if '''Z''' is replaced by a polynomial ring over a field ''F'' and '''Q''' is replaced by a [[field of rational functions]] over ''F'' in the same variables, with the only difference that "up to a sign" must be replaced by "up to the multiplication by an invertible constant in ''F''". This allows to reduce the factorization over a [[purely transcendental]] field extension of ''F'' to the factorization of [[multivariate polynomial]]s over ''F''.
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| ==Square-free factorization==
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| {{Main|square-free polynomial}}
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| If two or more factors of a polynomial are identical to each other, then the polynomial is a multiple of the square of this factor. In the case of univariate polynomials, this results in [[Multiplicity (mathematics)#Multiplicity of a root of a polynomial|multiple roots]]. In this case, then the multiple factor is also a factor of the polynomial's [[formal derivative|derivative]] (with respect to any of the variables, if several), which itself is a polynomial of lower degree. In the case of univariate polynomials over the rationals (or more generally over a field of [[characteristic (algebra)|characteristic]] zero), [[Square-free polynomial#Yun's algorithm|Yun's algorithm]] exploits this remark to factorize efficiently the polynomial into factors that are not multiple of a square and are therefore called '''square-free'''. To factorize the initial polynomial, it suffices to factorize each square-free factors. This algorithm is therefore the first step of almost all polynomial factorization algorithms.
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| Yun's algorithm extends easily to the multivariate case by considering a multivariate polynomial as an univariate polynomial over a polynomial ring.
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| In the case of a polynomial over a finite field, Yun's algorithm applies only if the degree is smaller than the characteristic, because, otherwise, the derivative of a non zero polynomial may be zero (over the field with ''p'' elements, the derivative of a polynomial in ''x''<sup>''p''</sup> is always zero). Nevertheless a succession of GCD computations, starting from the polynomial and its derivative, allows to compute the square-free decomposition; see [[Polynomial factorization over finite fields#Square-free factorization]].
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| Most factorization algorithms, including all the most efficient ones, begin by a square-free factorization.
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| ==Classical methods==
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| This section describes textbook methods that can be convenient when computing by hand. These methods are not used for computer computations because they use [[integer factorization]], which at the moment has a much higher complexity than polynomial factorization.
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| ===Obtaining linear factors===
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| All linear factors with rational coefficients can be found using the [[rational root test]]. If the polynomial to be factored is <math>a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0</math>, then all possible linear factors are of the form <math>b_1x-b_0</math>, where <math>b_1</math> is an integer factor of <math>a_n</math> and <math>b_0</math> is an integer factor of <math>a_0</math>. All possible combinations of integer factors can be tested for validity, and each valid one can be factored out using [[polynomial long division]]. If the original polynomial is the product of factors, at least two of which are of degree 2 or higher, this technique only provides a partial factorization; otherwise the factorization is complete. Note that in the case of a [[Cubic function|cubic polynomial]], if the cubic is factorisable at all, the rational root test gives a complete factorization, either into a linear factor and an irreducible quadratic factor, or into three linear factors.
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| ===Kronecker's method===
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| Since integer polynomials must factor into integer polynomial factors, and evaluating integer polynomials at integer values must produce integers, the integer values of a polynomial can be factored in only a finite number of ways, and produce only a finite number of possible polynomial factors.
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| For example, consider
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| :<math>f(x) = x^5 + x^4 + x^2 + x + 2</math>.
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| If this polynomial factors over '''Z''', then at least one of its factors must be of degree two or less. We need three values to uniquely fit a second degree polynomial. We'll use <math>f(0) = 2</math>, <math>f(1) = 6</math> and <math>f(-1) = 2</math>. Note that if one of those values were 0 then you already found a root (and so a factor). If none is 0, then each one has a finite amount of divisors. Now, 2 can only factor as
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| :1×2, 2×1, (−1)×(−2), or (−2)×(−1).
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| Therefore, if a second degree integer polynomial factor exists, it must take one of the values
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| :1, 2, −1, or −2
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| at <math>x=0</math>, and likewise at <math>x=-1</math>. There are eight different ways to factor 6 (one for each divisor of 6), so there are
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| :4×4×8 = 128
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| possible combinations, of which half can be discarded as the negatives of the other half, corresponding to 64 possible second degree integer polynomials that must be checked. These are the only possible integer polynomial factors of <math>f(x)</math>. Testing them exhaustively reveals that
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| :<math>p(x) = x^2 + x + 1</math>
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| constructed from <math>p(0)=1</math>, <math>p(1)=3</math> and <math>p(-1)=1</math>, factors <math>f(x)</math>.
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| Dividing <math>f</math> by <math>p</math> gives the other factor <math>q(x) = x^3 - x + 2</math>, so that <math>f = pq</math>.
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| Now one can test recursively to find factors of <math>p</math> and <math>q</math>. It turns out they both are irreducible over the integers, so that the irreducible factorization of <math>f</math> is
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| :<math>f(x) = p(x)q(x) = (x^2 + x + 1)(x^3 - x + 2) </math>
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| (''Van der Waerden'', Sections 5.4 and 5.6)
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| ==Modern methods==
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| ===Lifting modular factorization to factorization over the integers===
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| {{Expand section|date=September 2012}}
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| ===Lifting univariate factorization to multivariate factorization===
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| {{Expand section|date=September 2012}}
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| ===Factoring over finite fields===
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| {{Main|Factorization of polynomials over finite fields|Berlekamp's algorithm|Cantor–Zassenhaus algorithm}}
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| {{Expand section|date=September 2012}}
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| === Uses of LLL algorithm===
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| The first [[polynomial time]] algorithm for factoring rational polynomials has been discovered by Lenstra, Lenstra and Lovász and is an application of [[Lenstra–Lenstra–Lovász lattice basis reduction algorithm]], usually called "LLL algorithm". {{harv|Lenstra|Lenstra|Lovász|1982}} Although theoretically faster in the worst case, their factorization algorithm is not efficient in practice and is not used on computers.
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| However LLL algorithm is used by the fastest factorization algorithm to lift a modular factorization to a factorization over the integers.
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| One variation of LLL factorization algorithm runs as follows: calculate a complex (or ''p''-adic) root α of the polynomial ''P'' to high precision, then use the [[Lenstra–Lenstra–Lovász lattice basis reduction algorithm]] to find an approximate linear relation between 1, α, α<sup>2</sup>, α<sup>3</sup>, ... with integer coefficients, which with luck, is an exact linear relation and a polynomial factor of ''P''. One can determine a bound for the precision that guarantees that this method produces either a factor, or an irreducibility proof.
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| ===Factoring over algebraic extensions (Trager's method)===
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| We can factor a polynomial <math>p(x) \in K[x] </math>, where <math>K</math> is a finite field extension of <math>\mathbb{Q}</math>. First, using [[#Square-free factorization|square-free factorization]], we may suppose that the polynomial is square-free. Next we write <math>L= K[x]/p(x)</math> explicitly as an algebra over <math>\mathbb{Q}</math>. We next pick a random element <math>\alpha \in L</math>. By the primitive element theorem, <math>\alpha</math> generates <math>L</math> over <math>\mathbb{Q}</math> with high probability. If this is the case, we can compute the minimal polynomial, <math>q(y)\in \mathbb{Q}[y]</math> of <math>\alpha</math> over <math>\mathbb{Q}</math>. Factoring
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| :<math>q(y) = \prod_{i=1}^{n} q_i(y)</math>
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| over <math>\mathbb{Q}[y]</math>, we determine that
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| :<math>L = \mathbb{Q}[\alpha] = \mathbb{Q}[y]/q(y) = \prod_{i=1}^n \mathbb{Q}[y]/q_i(y)</math>
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| (notice that <math>L</math> is a [[reduced ring]] since <math>p(x)</math> is square-free), where <math>\alpha</math> corresponds to the element <math>(y,y,\ldots,y)</math>. Note that this is the unique decomposition of <math>L</math> as a product fields. Hence this decomposition is the same as
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| :<math>\prod_{i=1}^m K[x]/p_i(x)</math>
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| where
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| :<math>p(x) = \prod_{i=1}^m p_i(x)</math>
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| is the factorization of <math>p(x)</math> over <math>K[x]</math>. By writing <math>x\in L</math> and generators of <math>K</math> as a polynomials in <math>\alpha</math>, we can determine the embeddings of <math>x</math> and <math>K</math> into the components <math>\mathbb{Q}[y]/q_i(y)=K[x]/p_i(x)</math>. By finding the minimal polynomial of <math>x</math> in this ring, we have computed <math>p_i(x)</math>, and thus factored <math>p(x)</math> over <math>K.</math>
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| ==Bibliography==
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| {{Reflist}}
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| *{{citation |author1=Fröhlich, A.|author2=Shepherson, J. C.|title = On the factorisation of polynomials in a finite number of steps|journal = Mathematische Zeitschrift|volume = 62|issue=1|year = 1955|issn = 0025-5874|doi=10.1007/BF01180640}}
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| *{{citation |author1=Trager, B.M.|title = Algebraic Factoring and Rational Function Integration|journal = Proc. SYMSAC 76 http://dl.acm.org/citation.cfm?id=806338}}
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| * {{Cite journal|title=Quantitative Estimates for Polynomials in One or Several Variables: From Analysis and Number Theory to Symbolic and Massively Parallel Computation
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| |author=Bernard Beauzamy, [[Per Enflo]], Paul Wang
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| |journal=Mathematics Magazine
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| |volume=67
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| |issue=4
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| |date=October 1994
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| |pages=243–257
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| |jstor=2690843|ref=harv
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| |doi=10.2307/2690843}} (accessible to readers with undergraduate mathematics)
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| *{{Cite book | last1=Cohen | first1=Henri | title=A course in computational algebraic number theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-3-540-55640-4 | mr=1228206 | year=1993 | volume=138 | ref=harv | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->}}
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| * {{Citation|first1=Erich|last1=Kaltofen|chapter=Factorization of polynomials|title=Computer Algebra|publisher=Springer Verlag|year=1982|editor1 =B. Buchberger|editor2=R. Loos|editor3=G. Collins|url=http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.39.7916&rep=rep1&type=pdf|accessdate=September 20, 2012}}
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| *{{cite book
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| |author=[[Donald E. Knuth|Knuth, Donald E]]
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| |chapter=4.6.2 Factorization of Polynomials
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| |title=Seminumerical Algorithms
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| |series=The Art of Computer Programming
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| |volume=2
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| |edition=Third
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| |location=Reading, Massachusetts
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| |publisher=Addison-Wesley
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| |year=1997
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| |pages=439–461, 678–691<!-- xiv+762 -->
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| |isbn=0-201-89684-2}}
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| *{{Cite journal | last1=Lenstra | first1=A. K. | author1-link=A. K. Lenstra | last2=Lenstra | first2=H. W. | last3=Lovász | first3=László | author3-link=László Lovász | title=Factoring polynomials with rational coefficients | doi=10.1007/BF01457454 | mr=682664 | year=1982 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=261 | issue=4 | pages=515–534 | ref=harv | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->}}
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| * [[Bartel Leendert van der Waerden|Van der Waerden]], ''Algebra'' (1970), trans. Blum and Schulenberger, Frederick Ungar.
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| ==Further reading==
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| * {{Citation|first1=Erich|last1=Kaltofen|chapter=Polynomial Factorization 1982-1986|title=Computers in Mathematics|series=Lecture Notes in Pure and Applied Mathematics|volume=125|publisher=Marcel Dekker, Inc.|year=1990|editor1 =D. V. Chudnovsky|editor2=R. D. Jenks|url=http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.68.7461&rep=rep1&type=pdf|accessdate=October 14, 2012}}
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| * {{Citation|first1=Erich|last1=Kaltofen|chapter=Polynomial Factorization 1987–1991|title=Proceedings of Latin ’92|series=Springer Lect. Notes Comput. Sci.|volume=583|publisher=Springer|year=1992 |url=http://www4.ncsu.edu/~kaltofen/bibliography/92/Ka92_latin.pdf|accessdate=October 14, 2012}}
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| {{DEFAULTSORT:Factorization Of Polynomials}}
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| [[Category:Articles with inconsistent citation formats]]
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| [[Category:Polynomials]]
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| [[Category:Computer algebra]]
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