Editing Polynomial ring (section)

==Definition (multivariate case)==
{{Anchor|multivariable}}
Given {{mvar|n}} symbols <math>X_1, \dots, X_n,</math> called [[indeterminate (variable)|indeterminates]], a [[monomial]] (also called ''power product'') 
:<math>X_1^{\alpha_1}\cdots X_n^{\alpha_n}</math>
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, <math>X_1^0\cdots X_n^0 =1.</math>

The [[tuple]] of exponents {{math|1=''α'' = (''α''<sub>1</sub>, …, ''α''<sub>''n''</sub>)}} is called the ''multidegree'' or ''exponent vector'' of the monomial. For a less cumbersome notation, the abbreviation
:<math>X^\alpha=X_1^{\alpha_1}\cdots X_n^{\alpha_n}</math>
is often used. The ''degree'' of a monomial {{math|''X''<sup>''α''</sup>}}, frequently denoted {{math|deg ''α''}} or {{math|{{abs|''α''}}}}, is the sum of its exponents:
:<math> \deg \alpha = \sum_{i=1}^n \alpha_i. </math>

A ''polynomial'' in these indeterminates, with coefficients in a field {{mvar|K}}, or more generally a [[ring (mathematics)|ring]], is a finite [[linear combination]] of monomials
:<math> p = \sum_\alpha p_\alpha X^\alpha</math>
with coefficients in {{mvar|K}}. The ''degree'' of a nonzero polynomial is the maximum of the degrees of its monomials with nonzero coefficients.

The set of polynomials in <math>X_1, \dots, X_n,</math> denoted <math>K[X_1,\dots, X_n],</math> is thus a [[vector space]] (or a [[free module]], if {{mvar|K}} is a ring) that has the monomials as a basis.

<math>K[X_1,\dots, X_n]</math> is naturally equipped (see below) with a multiplication that makes a [[ring (mathematics)|ring]], and an [[associative algebra]] over {{mvar|K}}, called ''the polynomial ring in {{mvar|n}} indeterminates'' over {{mvar|K}} (the definite article ''the'' reflects that it is uniquely defined up to the name and the order of the indeterminates. If the ring {{mvar|K}} is [[commutative ring|commutative]], <math>K[X_1,\dots, X_n]</math> is also a commutative ring.

===Operations in {{math|''K''[''X''{{sub|1}}, ..., ''X''{{sub|''n''}}]}}===

''Addition'' and ''scalar multiplication'' of polynomials are those of a [[vector space]] or [[free module]] equipped by a specific basis (here the basis of the monomials). Explicitly, let
<math>p=\sum_{\alpha\in I}p_\alpha X^\alpha,\quad q=\sum_{\beta\in J}q_\beta X^\beta,</math>
where {{mvar|I}} and {{mvar|J}} are finite sets of exponent vectors.

The scalar multiplication of {{mvar|p}} and a scalar <math>c\in K</math> is
:<math>cp = \sum_{\alpha\in I}cp_\alpha X^\alpha.</math>

The addition of {{mvar|p}} and {{mvar|q}} is
:<math>p+q = \sum_{\alpha\in I\cup J}(p_\alpha+q_\alpha) X^\alpha,</math>
where <math>p_\alpha=0</math> if <math>\alpha \not\in I,</math> and <math>q_\beta=0</math> if <math>\beta \not\in J.</math> Moreover, if one has <math>p_\alpha+q_\alpha=0</math> for some <math>\alpha \in I \cap J,</math> the corresponding zero term is removed from the result.

The multiplication is 
:<math>pq = \sum_{\gamma\in I+J}\left(\sum_{\alpha, \beta\mid \alpha+\beta=\gamma} p_\alpha q_\beta\right) X^\gamma,</math>
where <math>I+J</math> is the set of the sums of one exponent vector in {{mvar|I}} and one other in {{mvar|J}} (usual sum of vectors). In particular, the product of two monomials is a monomial whose exponent vector is the sum of the exponent vectors of the factors.

The verification of the axioms of an [[associative algebra]] is straightforward.

===Polynomial expression===
{{main|Algebraic expression}}
{{Unreferenced section|date=January 2021}}
A '''polynomial expression''' is an [[expression (mathematics)|expression]] built with scalars (elements of {{mvar|K}}), indeterminates, and the operators of addition, multiplication, and exponentiation to nonnegative integer powers.

As all these operations are defined in <math>K[X_1,\dots, X_n]</math> a polynomial expression represents a polynomial, that is an element of <math>K[X_1,\dots, X_n].</math> The definition of a polynomial as a linear combination of monomials is a particular polynomial expression, which is often called the ''canonical form'', ''normal form'', or ''expanded form'' of the polynomial. 
Given a polynomial expression, one can compute the ''expanded'' form of the represented polynomial by ''expanding'' with the [[distributive law]] all the products that have a sum among their factors, and then using [[commutativity]] (except for the product of two scalars), and [[associativity]] for transforming the terms of the resulting sum into products of a scalar and a monomial; then one gets the canonical form by regrouping the [[like terms]].

The distinction between a polynomial expression and the polynomial that it represents is relatively recent, and mainly motivated by the rise of [[computer algebra]], where, for example, the test whether two polynomial expressions represent the same polynomial may be a nontrivial computation.

=== Categorical characterization ===
{{anchor|free commutative algebra|free commutative ring}}
If {{mvar|K}} is a commutative ring, the polynomial ring {{math|''K''[''X''<sub>1</sub>, …, ''X''<sub>''n''</sub>]}} has the following [[universal property]]: for every [[commutative algebra (structure)|commutative {{mvar|K}}-algebra]] {{mvar|A}}, and every {{mvar|n}}-[[tuple]] {{math|(''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>)}} of elements of {{mvar|A}}, there is a unique [[algebra homomorphism]] from {{math|''K''[''X''<sub>1</sub>, …, ''X''<sub>''n''</sub>]}} to {{mvar|A}} that maps each <math>X_i</math> to the corresponding <math>x_i.</math> This homomorphism is the ''evaluation homomorphism'' that consists in substituting <math>X_i</math> with <math>x_i</math> in every polynomial.

As it is the case for every universal property, this characterizes the pair <math>(K[X_1, \dots, X_n], (X_1, \dots, X_n))</math> up to a unique [[isomorphism]].

This may also be interpreted in terms of [[adjoint functor]]s. More precisely, let {{math|SET}} and {{math|ALG}} be respectively the [[category (mathematics)|categories]] of sets and commutative {{mvar|K}}-algebras (here, and in the following, the morphisms are trivially defined). There is a [[forgetful functor]] <math>\mathrm F: \mathrm{ALG}\to \mathrm{SET}</math> that maps algebras to their underlying sets. On the other hand, the map <math>X\mapsto K[X]</math> defines a functor <math>\mathrm{POL}: \mathrm{SET}\to \mathrm{ALG}</math> in the other direction. (If {{mvar|X}} is infinite, {{math|''K''[''X'']}} is the set of all polynomials in a finite number of elements of {{mvar|X}}.)

The universal property of the polynomial ring means that {{math|F}} and {{math|POL}} are [[adjoint functors]]. That is, there is a bijection
:<math>\operatorname{Hom}_{\mathrm {SET}}(X,\operatorname{F}(A))\cong \operatorname{Hom}_{\mathrm {ALG}}(K[X], A). </math>

This may be expressed also by saying that polynomial rings are '''free commutative algebras''', since they are [[free object]]s in the category of commutative algebras. Similarly, a polynomial ring with integer coefficients is the '''free commutative ring''' over its set of variables, since commutative rings and commutative algebras over the integers are the same thing.