In

below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Fred Below (1926â€“1988), American blues drummer
*Fritz von Below (1853â ...

).

^{''p''} is always zero). Nevertheless, a succession of GCD computations, starting from the polynomial and its derivative, allows one to compute the square-free decomposition; see Polynomial factorization over finite fields#Square-free factorization.

^{2}, Î±^{3}, . . . with integer coefficients, which might be an exact linear relation and a polynomial factor of $f(x)$. One can determine a bound for the precision that guarantees that this method produces either a factor, or an irreducibility proof. Although this method finishes in polynomial time, it is not used in practice because the lattice has high dimension and huge entries, which makes the computation slow.
The exponential complexity in the Zassenhaus algorithm comes from a combinatorial problem: how to select the right subsets of $f\_1(x),...,f\_r(x)$. State-of-the-art factoring implementations work in a manner similar to Zassenhaus, except that the combinatorial problem is translated to a lattice problem that is then solved by LLL.M. van Hoeij

''Factoring polynomials and the knapsack problem.''

Journal of Number Theory, 95, 167-189, (2002). In this approach, LLL is not used to compute coefficients of factors, but rather to compute vectors with $r$ entries in that encode the subsets of $f\_1(x),...,f\_r(x)$ corresponding to the irreducible true factors.

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

and computer algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, factorization of polynomials or polynomial factorization expresses a polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

with coefficients in a given field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

or in the integers
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of computer algebra system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software
Mathematical software is software used to mathematical model, model, analyze or calculate numeric, symbolic or geometric data.
It is a type of applica ...

s.
The first polynomial factorization algorithm was published by Theodor von SchubertFriedrich Theodor von Schubert (30 October 1758 – 21 October 1825) was a German astronomer and geographer. Born in Helmstedt, his father, Johann Ernst Schubert, was a professor of theology and abbot of Michaelstein Abbey. Theodor likewise studi ...

in 1793. Leopold Kronecker
Leopold Kronecker (; 7 December 1823 â€“ 29 December 1891) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

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
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software
Mathematical software is software used to mathematical model, model, analyze or calculate numeric, symbolic or geometric data.
It is a type of applicat ...

:
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. (Erich Kaltofen, 1982)Nowadays, modern algorithms and computers can quickly factor univariate polynomials of degree more than 1000 having coefficients with thousands of digits. For this purpose, even for factoring over the

rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s and number field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s, a fundamental step is a factorization of a polynomial over a finite field.
Formulation of the question

Polynomial ring
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s over the integers or over a field are unique factorization domain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s. This means that every element of these rings is a product of a constant and a product of irreducible polynomial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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.
Factorization depends on the base field. For example, the fundamental theorem of algebra
The fundamental theorem of algebra also known as d'Alembert's theorem or the d'Alembert-Gauss theorem states that every non-constant
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics)
In mathematics, the word constan ...

, which states that every polynomial with complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

coefficients has complex roots, implies that a polynomial with integer coefficients can be factored (with root-finding algorithms
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

) into s over the complex field C. Similarly, over the field of reals
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, the irreducible factors have degree at most two, while there are polynomials of any degree that are irreducible over the field of rationals
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

Q.
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. However, this is not a sufficient condition: FrÃ¶hlich and Shepherdson give examples of such fields for which no factorization algorithm can exist.
The fields of coefficients for which factorization algorithms are known include prime field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s (i.e., the field of rationals
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

and prime modular arithmetic #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure ( ...

) and their finitely generated field extension
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s. Integer coefficients are also tractable. Kronecker's classical method is interesting only from a historical point of view; modern algorithms proceed by a succession of:
* Square-free factorization
* Factorization over finite fields
and reductions:
* From the multivariate case to the univariate In mathematics, a univariate object is an expression, equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contai ...

case.
* From coefficients in a purely transcendental extension
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

to the multivariate case over the ground field (see below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Fred Below (1926â€“1988), American blues drummer
*Fritz von Below (1853â ...

).
* From coefficients in an algebraic extension to coefficients in the ground field (see below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Fred Below (1926â€“1988), American blues drummer
*Fritz von Below (1853â ...

).
* From rational coefficients to integer coefficients (see below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Fred Below (1926â€“1988), American blues drummer
*Fritz von Below (1853â ...

).
* From integer coefficients to coefficients in a prime field with ''p'' elements, for a well chosen ''p'' (see Primitive partâ€“content factorization

In this section, we show that factoring over Q (the rational numbers) and over Z (the integers) is essentially the same problem. The ''content'' of a polynomial ''p'' âˆˆ Z 'X'' denoted "cont(''p'')", is,up to Two mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

its sign, the greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

of its coefficients. The ''primitive part'' of ''p'' is primpart(''p'')=''p''/cont(''p''), which is a 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.
For example,
:$-10x^2\; +\; 5x\; +\; 5\; =\; (-5)\; (2x^2\; -\; x\; -\; 1)\; \backslash ,$
is a factorization into content and primitive part.
Every polynomial ''q'' with rational coefficients may be written
:$q\; =\; \backslash frac,$
where ''p'' âˆˆ Z 'X''and ''c'' âˆˆ 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:
:$\backslash text\; (q)\; =\backslash frac,$
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.
For example,
:$\backslash frac\; +\; \backslash frac\; +\; 2x\; +\; 1\; =\; \backslash frac$
is a factorization into content and primitive part.
Gauss
Johann Carl Friedrich Gauss (; german: GauÃŸ ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of m ...

proved that the product of two primitive polynomials is also primitive ( 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. Similarly, 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.
In other words, an integer GCD computation reduces the factorization of a polynomial over the rationals to the factorization of a primitive polynomial with integer coefficients, and the factorization over the integers to the factorization of an integer and a primitive polynomial.
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
In abstract algebra, the field of fractions of an integral domain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space ...

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 reduces the factorization over a purely transcendental
In abstract algebra, the transcendence degree of a field extension ''L'' /''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an Algebraic independence, algebraically i ...

field extension of ''F'' to the factorization of multivariate polynomial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s over ''F''.
Square-free factorization

If two or more factors of a polynomial are identical, then the polynomial is a multiple of the square of this factor. The multiple factor is also a factor of the polynomial'sderivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

(with respect to any of the variables, if several).
For univariate polynomials, multiple factors are equivalent to multiple roots (over a suitable extension field). For univariate polynomials over the rationals (or more generally over a field of characteristic
Characteristic (from the Greek word for a property, attribute or trait
Trait may refer to:
* Phenotypic trait in biology, which involve genes and characteristics of organisms
* Trait (computer programming), a model for structuring object-oriented ...

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 computations starting with gcd(''f''(''x''), ''f'' '(''x'')). To factorize the initial polynomial, it suffices to factorize each square-free factor. Square-free factorization is therefore the first step in most polynomial factorization algorithms.
Yun's algorithm extends this to the multivariate case by considering a multivariate polynomial as a univariate polynomial over a polynomial ring.
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''Classical methods

This section describes textbook methods that can be convenient when computing by hand. These methods are not used for computer computations because they useinteger factorization
In number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777â€“1855) said, "Mathematics is the queen of the sciencesâ€”and number ...

, which is currently slower than polynomial factorization.
The two methods that follow start from a univariate polynomial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

with integer coefficients for finding factors that are also polynomials with integer coefficients.
Obtaining linear factors

All linear factors withrational
Rationality is the quality or state of being rational â€“ that is, being based on or agreeable to reason
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:Î»Î¿Î³Î¹ ...

coefficients can be found using the rational root test
In algebra
Algebra (from ar, Ø§Ù„Ø¬Ø¨Ø±, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...

. If the polynomial to be factored is $a\_nx^n\; +\; a\_x^\; +\; \backslash cdots\; +\; a\_1x\; +\; a\_0$, then all possible linear factors are of the form $b\_1x-b\_0$, where $b\_1$ is an integer factor of $a\_n$ and $b\_0$ is an integer factor of $a\_0$. All possible combinations of integer factors can be tested for validity, and each valid one can be factored out using polynomial long division
In algebra
Algebra (from ar, Ø§Ù„Ø¬Ø¨Ø±, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...

. 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. In particular, if there is exactly one non-linear factor, it will be the polynomial left after all linear factors have been factorized out. In the case of a cubic polynomial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, if the cubic is factorizable 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.
Kronecker's method

Kronecker's method is aimed to factorunivariate polynomial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s with integer coefficients into polynomials with integer coefficients.
The method uses the fact that evaluating integer polynomials at integer values must produce integers. That is, if $f(x)$ is a polynomial with integer coefficients, then $f(a)$ is an integer as soon as is an integer. There are only a finite number of possible integer values for a factor of . So, if $g(x)$ is a factor of $f(x),$ the value of $g(a)$ must be one of the factors of $f(a).$
If one searches the factors of a given degree , one can consider $d+1$ values, $a\_0,\; \backslash ldots,\; a\_d$ for , which give a finite number of possibilities for the tuple
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

$(f(a\_0),\backslash ldots,\; f(a\_d).$ Each such tuple defines a unique polynomial of degree at most , which can be computed by polynomial interpolation
In numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathemati ...

, and tested for being a factor by polynomial division
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower Degree of a polynomial, degree, a generalized version of the familiar arithmetic technique called long division. It can be don ...

. So, an exhaustive search allows finding all factors of degree at most .
For example, consider
:$f(x)\; =\; x^5\; +\; x^4\; +\; x^2\; +\; x\; +\; 2$.
If this polynomial factors over Z, then at least one of its factors $p(x)$ must be of degree two or less, so $p(x)$ is uniquely . Thus, we compute three values $f(0)\; =\; 2$, $f(1)\; =\; 6$ and $f(-1)\; =\; 2$. If one of these values is 0, we have a linear factor. If the values are nonzero, we can list the possible factorizations for each. Now, 2 can only factor as
:1Ã—2, 2Ã—1, (âˆ’1)Ã—(âˆ’2), or (âˆ’2)Ã—(âˆ’1).
Therefore, if a second degree integer polynomial factor exists, it must take one of the values
:''p''(0) ''='' 1, 2, âˆ’1, or âˆ’2
and likewise for ''p''(1). There are eight factorizations of 6 (four each for 1Ã—6 and 2Ã—3), making a total of 4Ã—4Ã—8 = 128 possible triples (''p''(0), ''p''(1), ''p''(âˆ’1)), of which half can be discarded as the negatives of the other half. Thus, we must check 64 explicit integer polynomials $p(x)\; =\; ax^2+bx+c$ as possible factors of $f(x)$. Testing them exhaustively reveals that
:$p(x)\; =\; x^2\; +\; x\; +\; 1$
constructed from (''g''(0), ''g''(1), ''g''(âˆ’1)) = (1,3,1) factors $f(x)$.
Dividing ''f''(''x'') by ''p''(''x'') gives the other factor $q(x)\; =\; x^3\; -\; x\; +\; 2$, so that $f(x)\; =\; p(x)q(x)$.
Now one can test recursively to find factors of ''p''(''x'') and ''q''(''x''), in this case using the rational root test. It turns out they are both irreducible, so the irreducible factorization of ''f''(''x'') is:
:$f(x)\; =\; p(x)q(x)\; =\; (x^2\; +\; x\; +\; 1)(x^3\; -\; x\; +\; 2).$
Modern methods

Factoring over finite fields

Factoring univariate polynomials over the integers

If $f(x)$ is a univariate polynomial over the integers, assumed to be content-free andsquare-free {{no footnotes, date=December 2015
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

, one starts by computing a bound $B$
such that any factor $g(x)$ has coefficients of
absolute value bounded by $B$. This way, if $m$ is
an integer larger than $2B$, and if $g(x)$ is known modulo
$m$, then $g(x)$ can be reconstructed from its image mod $m$.
The Zassenhaus algorithm proceeds as follows. First, choose a prime
number $p$ such that the image of $f(x)$ mod $p$
remains square-free {{no footnotes, date=December 2015
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

, and of the same degree as $f(x)$.
Then factor $f(x)$ mod $p$. This produces integer polynomials $f\_1(x),...,f\_r(x)$ whose product matches $f(x)$ mod $p$. Next, apply Hensel lifting; this updates the $f\_i(x)$ in such a way that their product matches $f(x)$ mod $p^a$, where $a$ is large enough that $p^a$ exceeds $2B$: thus each $f\_i(x)$ corresponds to a well-defined integer polynomial. Modulo $p^a$, the polynomial $f(x)$ has $2^r$ factors (up to units): the products of all subsets of $$ mod $p^a$. These factors modulo $p^a$ need not correspond to "true" factors of $f(x)$ in $\backslash mathbb\; Z;\; href="/html/ALL/s/.html"\; ;"title="">$polynomial time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by th ...

algorithm for factoring rational polynomials was discovered by Lenstra, Lenstra and LovÃ¡sz and is an application of the Lenstraâ€“Lenstraâ€“LovÃ¡sz lattice basis reduction (LLL) algorithm .
A simplified version of the LLL factorization algorithm is as follows: calculate a complex (or ''p''-adic) root Î± of the polynomial $f(x)$ to high precision, then use the Lenstraâ€“Lenstraâ€“LovÃ¡sz lattice basis reduction algorithm to find an approximate linear relation
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

between 1, Î±, Î±''Factoring polynomials and the knapsack problem.''

Journal of Number Theory, 95, 167-189, (2002). In this approach, LLL is not used to compute coefficients of factors, but rather to compute vectors with $r$ entries in that encode the subsets of $f\_1(x),...,f\_r(x)$ corresponding to the irreducible true factors.

Factoring over algebraic extensions (Trager's method)

We can factor a polynomial $p(x)\; \backslash in\; K;\; href="/html/ALL/s/.html"\; ;"title="">$, where the field $K$ is a finite extension of $\backslash mathbb$. First, usingsquare-free factorization
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, we may suppose that the polynomial is square-free. Next we define the quotient ring
In ring theory
In algebra, ring theory is the study of ring (mathematics), ringsâ€”algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...

$L=\; K;\; href="/html/ALL/s/.html"\; ;"title="">$ of degree $n=;\; href="/html/ALL/s/:\backslash mathbb.html"\; ;"title=":\backslash mathbb">:\backslash mathbb$reduced ringIn ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, ''x''2 = 0 implies ''x'' = 0. A commutative ...

since $p(x)$ is square-free. Indeed, if$p(x)\; =\; \backslash prod\_^m\; p\_i(x)$is the desired factorization of ''p''(''x''), the ring decomposes uniquely into fields as: :$L\; =\; K;\; href="/html/ALL/s/.html"\; ;"title="">$ We will find this decomposition without knowing the factorization. First, we write ''L'' explicitly as an algebra over $\backslash mathbb$: we pick a random element $\backslash alpha\; \backslash in\; L$, which generates $L$ over $\backslash mathbb$ with high probability by the

primitive element theorem In field theory, the primitive element theorem is a result characterizing the finite degree field extension
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas an ...

. If this is the case, we can compute the minimal polynomial $q(y)\backslash in\; \backslash mathbb;\; href="/html/ALL/s/.html"\; ;"title="">$See also

* , for elementary heuristic methods and explicit formulasBibliography

* * * (accessible to readers with undergraduate mathematics) * * * * * Van der Waerden, ''Algebra'' (1970), trans. Blum and Schulenberger, Frederick Ungar.Further reading

* * * {{Polynomials Polynomials Computer algebra