Editing Polynomial ring (section)

===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.