In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in the field of
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, a polynomial ring or polynomial algebra is a
ring (which is also a
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
) formed from the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of
polynomial
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 positive-integer powers of variables. An exampl ...
s in one or more
indeterminates (traditionally also called
variables) with
coefficients in another
ring, often a
field.
Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the
integers.
Polynomial rings occur and are often fundamental in many parts of mathematics such as
number theory,
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, and
algebraic geometry. In
ring theory, many classes of rings, such as
unique factorization domains,
regular rings,
group rings,
rings of formal power series,
Ore polynomials,
graded rings, have been introduced for generalizing some properties of polynomial rings.
A closely related notion is that of the
ring of polynomial functions In mathematics, the ring of polynomial functions on a vector space ''V'' over a field ''k'' gives a coordinate-free analog of a polynomial ring. It is denoted by ''k'' 'V'' If ''V'' is finite dimensional and is viewed as an algebraic variety, th ...
on a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, and, more generally,
ring of regular functions on an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
.
Definition (univariate case)
The polynomial ring, , in over a
field (or, more generally, a
commutative ring) can be defined in several equivalent ways. One of them is to define as the set of expressions, called polynomials in , of the form
:
where , the coefficients of , are elements of , if , and are symbols, which are considered as "powers" of , and follow the usual rules of
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
: , , and
for any
nonnegative integers and . The symbol is called an indeterminate or variable. (The term of "variable" comes from the terminology of
polynomial functions. However, here, has not any value (other than itself), and cannot vary, being a ''constant'' in the polynomial ring.)
Two polynomials are equal when the corresponding coefficients of each are equal.
One can think of the ring as arising from by adding one new element that is external to , commutes with all elements of , and has no other specific properties. This can be used for an equivalent definition of polynomial rings.
The polynomial ring in over is equipped with an addition, a multiplication and a
scalar multiplication that make it a
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
. These operations are defined according to the ordinary rules for manipulating algebraic expressions. Specifically, if
:
and
:
then
:
and
:
where ,
:
and
:
In these formulas, the polynomials and are extended by adding "dummy terms" with zero coefficients, so that all and that appear in the formulas are defined. Specifically, if , then for .
The scalar multiplication is the special case of the multiplication where is reduced to its ''constant term'' (the term that is independent of ); that is
:
It is straightforward to verify that these three operations satisfy the axioms of a commutative algebra over . Therefore, polynomial rings are also called ''polynomial algebras''.
Another equivalent definition is often preferred, although less intuitive, because it is easier to make it completely rigorous, which consists in defining a polynomial as an infinite
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of elements of , having the property that only a finite number of the elements are nonzero, or equivalently, a sequence for which there is some so that for . In this case, and are considered as alternate notations for
the sequences and , respectively. A straightforward use of the operation rules shows that the expression
:
is then an alternate notation for the sequence
:.
Terminology
Let
:
be a nonzero polynomial with
The ''constant term'' of is
It is zero in the case of the zero polynomial.
The ''degree'' of , written is
the largest such that the coefficient of is not zero.
The ''leading coefficient'' of is
In the special case of the zero polynomial, all of whose coefficients are zero, the leading coefficient is undefined, and the degree has been variously left undefined, defined to be , or defined to be a .
A ''constant polynomial'' is either the zero polynomial, or a polynomial of degree zero.
A nonzero polynomial is
monic if its leading coefficient is
Given two polynomials and , one has
:
and, over a
field, or more generally an
integral domain,
:
It follows immediately that, if is an integral domain, then so is .
It follows also that, if is an integral domain, a polynomial is a
unit (that is, it has a
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
) if and only if it is constant and is a unit in .
Two polynomials are
associated if either one is the product of the other by a unit.
Over a field, every nonzero polynomial is associated to a unique monic polynomial.
Given two polynomials, and , one says that ''divides'' , is a ''divisor'' of , or is a multiple of , if there is a polynomial such that .
A polynomial is
irreducible if it is not the product of two non-constant polynomials, or equivalently, if its divisors are either constant polynomials or have the same degree.
Polynomial evaluation
Let be a field or, more generally, a
commutative ring, and a ring containing . For any polynomial in and any element in , the substitution of with in defines an element of , which is
denoted . This element is obtained by carrying on in after the substitution the operations indicated by the expression of the polynomial. This computation is called the evaluation of at . For example, if we have
:
we have
:
(in the first example , and in the second one ). Substituting for itself results in
:
explaining why the sentences "Let be a polynomial" and "Let be a polynomial" are equivalent.
The ''polynomial function'' defined by a polynomial is the function from into that is defined by
If is an infinite field, two different polynomials define different polynomial functions, but this property is false for finite fields. For example, if is a field with elements, then the polynomials and both define the zero function.
For every in , the evaluation at , that is, the map
defines an
algebra homomorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in ,
* F(kx) = kF( ...
from to , which is the unique homomorphism from to that fixes , and maps to . In other words, has the following
universal property:
:''For every ring containing , and every element of , there is a unique algebra homomorphism from'' ''to that fixes , and maps to .''
The
image of the map
, that is, the subset of obtained by substituting for in elements of , is denoted . For example,
, where
.
As for all universal properties, this defines the pair up to a unique isomorphism, and can therefore be taken as a definition of .
Univariate polynomials over a field
If is a
field, the polynomial ring has many properties that are similar to those of the
ring of integers
Most of these similarities result from the similarity between the
long division of integers and the
long division of polynomials.
Most of the properties of that are listed in this section do not remain true if is not a field, or if one considers polynomials in several indeterminates.
Like for integers, the
Euclidean division of polynomials
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 concept is analogous to the greatest common ...
has a property of uniqueness. That is, given two polynomials and in , there is a unique pair of polynomials such that , and either or . This makes a
Euclidean domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers ...
. However, most other Euclidean domains (except integers) do not have any property of uniqueness for the division nor an easy algorithm (such as long division) for computing the Euclidean division.
The Euclidean division is the basis of the
Euclidean algorithm for polynomials that computes a
polynomial greatest common divisor
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 concept is analogous to the greatest common ...
of two polynomials. Here, "greatest" means "having a maximal degree" or, equivalently, being maximal for the
preorder defined by the degree. Given a greatest common divisor of two polynomials, the other greatest common divisors are obtained by multiplication by a nonzero constant (that is, all greatest common divisors of and are associated). In particular, two polynomials that are not both zero have a unique greatest common divisor that is monic (leading coefficient equal to ).
The
extended Euclidean algorithm allows computing (and proving)
Bézout's identity
In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout, is the following theorem:
Here the greatest common divisor of and is taken to be . The integers and are called Bézout coefficients for ; they ...
. In the case of , it may be stated as follows. Given two polynomials and of respective degrees and , if their monic greatest common divisor has the degree , then there is a unique pair of polynomials such that
:
and
:
(For making this true in the limiting case where or , one has to define as negative the degree of the zero polynomial. Moreover, the equality
can occur only if and are associated.) The uniqueness property is rather specific to . In the case of the integers the same property is true, if degrees are replaced by absolute values, but, for having uniqueness, one must require .
Euclid's lemma applies to . That is, if divides , and is
coprime with , then divides . Here, ''coprime'' means that the monic greatest common divisor is . ''Proof:'' By hypothesis and Bézout's identity, there are , , and such that and . So
The
unique factorization property results from Euclid's lemma. In the case of integers, this is the
fundamental theorem of arithmetic. In the case of , it may be stated as: ''every non-constant polynomial can be expressed in a unique way as the product of a constant, and one or several irreducible monic polynomials; this decomposition is unique up to the order of the factors.'' In other terms is a
unique factorization domain. If is the field of complex numbers, the
fundamental theorem of algebra asserts that a univariate polynomial is irreducible if and only if its degree is one. In this case the unique factorization property can be restated as: ''every non-constant univariate polynomial over the complex numbers can be expressed in a unique way as the product of a constant, and one or several polynomials of the form'' ; ''this decomposition is unique up to the order of the factors.'' For each factor, is a
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
of the polynomial, and the number of occurrences of a factor is the
multiplicity of the corresponding root.
Derivation
The
(formal) derivative of the polynomial
:
is the polynomial
:
In the case of polynomials with
real or
complex coefficients, this is the standard
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
. The above formula defines the derivative of a polynomial even if the coefficients belong to a ring on which no notion of
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
is defined. The derivative makes the polynomial ring a
differential algebra.
The existence of the derivative is one of the main properties of a polynomial ring that is not shared with integers, and makes some computations easier on a polynomial ring than on integers.
Square-free factorization
Lagrange interpolation
Polynomial decomposition
Factorization
Except for factorization, all previous properties of are
effective, since their proofs, as sketched above, are associated with
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s for testing the property and computing the polynomials whose existence are asserted. Moreover these algorithms are efficient, as their
computational complexity is a
quadratic function of the input size.
The situation is completely different for factorization: the proof of the unique factorization does not give any hint for a method for factorizing. Already for the integers, there is no known algorithm running on a classical computer for factorizing them in
polynomial time. This is the basis of the
RSA cryptosystem, widely used for secure Internet communications.
In the case of , the factors, and the methods for computing them, depend strongly on . Over the complex numbers, the irreducible factors (those that cannot be factorized further) are all of degree one, while, over the real numbers, there are irreducible polynomials of degree 2, and, over the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, there are irreducible polynomials of any degree. For example, the polynomial
is irreducible over the rational numbers, is factored as
over the real numbers and, and as
over the complex numbers.
The existence of a factorization algorithm depends also on the ground field. In the case of the real or complex numbers,
Abel–Ruffini theorem shows that the roots of some polynomials, and thus the irreducible factors, cannot be computed exactly. Therefore, a factorization algorithm can compute only approximations of the factors. Various algorithms have been designed for computing such approximations, see
Root finding of polynomials
In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex n ...
.
There is an example of a field such that there exist exact algorithms for the arithmetic operations of , but there cannot exist any algorithm for deciding whether a polynomial of the form
is
irreducible or is a product of polynomials of lower degree.
On the other hand, over the rational numbers and over finite fields, the situation is better than for
integer factorization, as there are
factorization algorithms that have a
polynomial complexity. They are implemented in most general purpose
computer algebra system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The ...
s.
Minimal polynomial
If is an element of an
associative -algebra , the
polynomial evaluation at is the unique
algebra homomorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in ,
* F(kx) = kF( ...
from into that maps to and does not affect the elements of itself (it is the
identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
on ). It consists of ''substituting'' with in every polynomial. That is,
:
The image of this ''evaluation homomorphism'' is the subalgebra generated by , which is necessarily commutative.
If is injective, the subalgebra generated by is isomorphic to . In this case, this subalgebra is often denoted by . The notation ambiguity is generally harmless, because of the isomorphism.
If the evaluation homomorphism is not injective, this means that its
kernel is a nonzero
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
, consisting of all polynomials that become zero when is substituted with . This ideal consists of all multiples of some monic polynomial, that is called the minimal polynomial of . The term ''minimal'' is motivated by the fact that its degree is minimal among the degrees of the elements of the ideal.
There are two main cases where minimal polynomials are considered.
In
field theory and
number theory, an element of an
extension field of is
algebraic
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
over if it is a root of some polynomial with coefficients in . The
minimal polynomial over of is thus the monic polynomial of minimal degree that has as a root. Because is a field, this minimal polynomial is necessarily
irreducible over . For example, the minimal polynomial (over the reals as well as over the rationals) of the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
is
. The
cyclotomic polynomials are the minimal polynomials of the
roots of unity.
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 through matrice ...
, the
square matrices over form an
associative -algebra of finite dimension (as a vector space). Therefore the evaluation homomorphism cannot be injective, and every matrix has a
minimal polynomial (not necessarily irreducible). By
Cayley–Hamilton theorem, the evaluation homomorphism maps to zero the
characteristic polynomial of a matrix. It follows that the minimal polynomial divides the characteristic polynomial, and therefore that the degree of the minimal polynomial is at most .
Quotient ring
In the case of , the
quotient ring by an ideal can be built, as in the general case, as a set of
equivalence classes. However, as each equivalence class contains exactly one polynomial of minimal degree, another construction is often more convenient.
Given a polynomial of degree , the ''quotient ring'' of by the
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
generated by can be identified with the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of the polynomials of degrees less than , with the "multiplication modulo " as a multiplication, the ''multiplication modulo'' consisting of the remainder under the division by of the (usual) product of polynomials. This quotient ring is variously denoted as
or simply
The ring
is a field if and only if is an
irreducible polynomial. In fact, if is irreducible, every nonzero polynomial of lower degree is coprime with , and
Bézout's identity
In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout, is the following theorem:
Here the greatest common divisor of and is taken to be . The integers and are called Bézout coefficients for ; they ...
allows computing and such that ; so, is the
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
of modulo . Conversely, if is reducible, then there exist polynomials of degrees lower than such that ; so are nonzero
zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
s modulo , and cannot be invertible.
For example, the standard definition of the field of the complex numbers can be summarized by saying that it is the quotient ring
:
and that the image of in
is denoted by . In fact, by the above description, this quotient consists of all polynomials of degree one in , which have the form , with and in
The remainder of the Euclidean division that is needed for multiplying two elements of the quotient ring is obtained by replacing by in their product as polynomials (this is exactly the usual definition of the product of complex numbers).
Let be an
algebraic element
In mathematics, if is a field extension of , then an element of is called an algebraic element over , or just algebraic over , if there exists some non-zero polynomial with coefficients in such that . Elements of which are not algebraic ove ...
in a -algebra . By ''algebraic'', one means that has a minimal polynomial . The
first ring isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exis ...
asserts that the substitution homomorphism induces an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
of
onto the image of the substitution homomorphism. In particular, if is a
simple extension of generated by , this allows identifying and
This identification is widely used in
algebraic number theory.
Modules
The
applies to
''K''
'X'' when ''K'' is a field. This means that every finitely generated module over ''K''
'X''may be decomposed into a
direct sum of a
free module and finitely many modules of the form
, where ''P'' is an
irreducible polynomial over ''K'' and ''k'' a positive integer.
Definition (multivariate case)
Given symbols
called
indeterminates, a
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
(also called ''power product'')
:
is a formal product of these indeterminates, possibly raised to a nonnegative power. As usual, exponents equal to one and factors with a zero exponent can be omitted. In particular,
The
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
of exponents is called the ''multidegree'' or ''exponent vector'' of the monomial. For a less cumbersome notation, the abbreviation
:
is often used. The ''degree'' of a monomial , frequently denoted or , is the sum of its exponents:
:
A ''polynomial'' in these indeterminates, with coefficients in a field, or more generally a
ring, is a finite
linear combination of monomials
:
with coefficients in . The ''degree'' of a nonzero polynomial is the maximum of the degrees of its monomials with nonzero coefficients.
The set of polynomials in
denoted
is thus a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
(or a
free module, if is a ring) that has the monomials as a basis.