Editing Polynomial ring (section)

=== Quotient ring===
In the case of {{math|''K''[''X'']}}, the [[quotient ring]] by an ideal can be built, as in the general case, as a set of [[equivalence class]]es. However, as each equivalence class contains exactly one polynomial of minimal degree, another construction is often more convenient.

Given a polynomial {{mvar|p}} of degree {{mvar|d}}, the ''quotient ring'' of {{math|''K''[''X'']}} by the [[ideal (ring theory)|ideal]] generated by {{mvar|p}} can be identified with the [[vector space]] of the polynomials of degrees less than {{mvar|d}}, with the "multiplication modulo {{mvar|p}}" as a multiplication, the ''multiplication  modulo'' {{mvar|p}} consisting of the remainder under the division by {{mvar|p}} of the   (usual) product of polynomials. This quotient ring is variously denoted as <math>K[X]/pK[X],</math> <math>K[X]/\langle p \rangle,</math> <math>K[X]/(p),</math> or simply <math>K[X]/p.</math>

The ring <math>K[X]/(p)</math> is a field if and only if {{mvar|p}} is an [[irreducible polynomial]]. In fact, if {{mvar|p}} is irreducible, every nonzero polynomial {{mvar|q}} of lower degree is coprime with {{mvar|p}}, and [[Bézout's identity]] allows computing {{mvar|r}} and {{mvar|s}} such that {{math|1=''sp'' + ''qr'' = 1}}; so, {{mvar|r}} is the [[multiplicative inverse]] of {{mvar|q}} modulo {{mvar|p}}. Conversely, if {{mvar|p}} is reducible, then there exist polynomials {{mvar|a, b}} of degrees lower than {{math|deg(''p'')}} such that {{math|1=''ab'' = ''p''}}&thinsp;; so {{mvar|a, b}} are nonzero [[zero divisor]]s modulo {{mvar|p}}, 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 
:<math>\mathbb C =\mathbb R[X]/(X^2+1),</math> 
and that the image of {{mvar|X}} in <math>\mathbb C</math> is denoted by {{mvar|i}}. In fact, by the above description, this quotient consists of all polynomials of degree one in {{mvar|i}}, which have the form {{math|''a'' + ''bi''}}, with {{mvar|a}} and {{mvar|b}} in <math>\mathbb R.</math> The remainder of the Euclidean division that is needed for multiplying two elements of the quotient ring is obtained by replacing {{math|''i''{{sup|2}}}} by {{math|−1}} in their product as polynomials (this is exactly the usual definition of the product of complex numbers).

Let {{math|''θ''}} be an [[algebraic element]] in a {{mvar|K}}-algebra {{mvar|A}}. By ''algebraic'', one means that {{math|''θ''}} has a minimal polynomial {{mvar|p}}. The [[first ring isomorphism theorem]] asserts that the substitution homomorphism induces an [[isomorphism]] of <math>K[X]/(p)</math> onto the image {{math|''K''[''θ'']}} of the substitution homomorphism. In particular, if {{mvar|A}} is a [[simple extension]] of {{mvar|K}} generated by {{math|''θ''}}, this allows identifying {{mvar|A}} and <math>K[X]/(p).</math> This identification is widely used in [[algebraic number theory]].