Polynomial ring

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates 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, 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 on a vector space, and, more generally, ring of regular functions on an algebraic variety.

Definition (univariate case)

Let be a field or a commutative ring.
The polynomial ring in over, which is denoted, 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 is a nonnegative integer, the coefficients of are elements of, and are symbols called "powers" of that follow the usual rules of exponents:,, and for any nonnegative integers and. The symbol is called an indeterminate or variable.
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. 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 ; 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 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 number 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, if the degree of the zero polynomial is defined to be 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 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
. 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 from to, which is the unique homomorphism from to that fixes, and maps to. In other words, has the following universal property:
As for all universal properties, this defines the pair up to a unique isomorphism, and can therefore be taken as a definition of.
The image of the map, that is, the subset of obtained by substituting for in elements of, is denoted and called the adjunction of a to K. For example,, and the simplification rules for the powers of a square root imply

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 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. However, most other Euclidean domains do not have any property of uniqueness for the division nor an easy algorithm for computing the Euclidean division.
The Euclidean division is the basis of the Euclidean algorithm for polynomials that computes a polynomial greatest common divisor 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. In particular, two polynomials that are not both zero have a unique greatest common divisor that is monic.
The extended Euclidean algorithm allows computing Bézout's identity. 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
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 of the polynomial, and the number of occurrences of a factor is the multiplicity of the corresponding root.

Derivation

The derivative of the polynomial
is the polynomial
In the case of polynomials with real or complex coefficients, this is the standard derivative. The above formula defines the derivative of a polynomial even if the coefficients belong to a ring on which no notion of limit 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

A polynomial with coefficients in a field or integral domain is square-free if it does not have a multiple root in the algebraically closed field containing its coefficients. In particular, a polynomial of degree with real or complex coefficients is square-free if it has distinct complex roots. Equivalently, a polynomial over a field is square-free if and only if the greatest common divisor of the polynomial and its derivative is.
A square-free factorization of a polynomial is an expression for that polynomial as a product of powers of pairwise relatively prime square-free factors. Over the real numbers, such a factorization can be computed efficiently by Yun's algorithm. Less efficient algorithms are known for square-free factorization of polynomials over finite fields.