Editing Polynomial ring (section)

===Minimal polynomial===
{{main|Minimal polynomial (field theory)}}

If {{math|''θ''}} is an element of an [[associative algebra|associative {{mvar|K}}-algebra]] {{math|''L''}}, the [[#Polynomial evaluation|polynomial evaluation]] at {{math|''θ''}} is the unique [[algebra homomorphism]] {{math|''φ''}} from {{math|''K''[''X'']}} into {{math|''L''}} that maps {{math|''X''}} to {{math|''θ''}} and does not affect the elements of {{math|''K''}} itself (it is the [[identity function|identity map]] on {{math|''K''}}). It consists of ''substituting'' {{math|''X''}} with {{math|''θ''}} in every polynomial. That is,  
: <math>
  \varphi\left(a_m X^m + a_{m - 1} X^{m - 1} + \cdots + a_1 X + a_0\right) = 
  a_m \theta^m + a_{m - 1} \theta^{m - 1} + \cdots + a_1 \theta + a_0.
</math>

The image of this ''evaluation homomorphism'' is the subalgebra generated by {{mvar|θ}}, which is necessarily commutative.
If {{math|''φ''}} is injective, the subalgebra generated by {{mvar|θ}} is isomorphic to {{math|''K''[''X'']}}. In this case, this subalgebra is often denoted by {{math|''K''[''θ'']}}. The notation ambiguity is generally harmless, because of the isomorphism.

{{anchor|minimal polynomial}}
If the evaluation homomorphism is not injective, this means that its [[kernel (algebra)|kernel]] is a nonzero [[ideal (ring theory)|ideal]], consisting of all polynomials that become zero when {{mvar|X}} is substituted with {{mvar|θ}}. This ideal consists of all multiples of some monic polynomial, that is called the '''minimal polynomial''' of {{mvar|θ}}. 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 (mathematics)|field theory]] and [[number theory]], an element {{mvar|θ}} of an [[extension field]] {{mvar|L}} of {{mvar|K}} is [[algebraic element|algebraic]] over {{mvar|K}} if it is  a root of some polynomial with coefficients in {{mvar|K}}. The [[minimal polynomial (field theory)|minimal polynomial]] over {{mvar|K}} of {{mvar|θ}} is thus the monic polynomial of minimal degree that has {{mvar|θ}} as a root. Because {{mvar|L}} is a field, this minimal polynomial is necessarily [[irreducible polynomial|irreducible]] over {{mvar|K}}. For example, the minimal polynomial (over the reals as well as over the rationals) of the [[complex number]] {{mvar|i}} is <math>X^2 + 1</math>. The [[cyclotomic polynomial]]s are the minimal polynomials of the [[roots of unity]].

In [[linear algebra]], the {{math|''n''×''n''}} [[square matrices]] over {{mvar|K}} form an [[associative algebra|associative {{mvar|K}}-algebra]] of finite dimension (as a vector space). Therefore the evaluation homomorphism cannot be injective, and every matrix has a [[minimal polynomial (linear algebra)|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 {{mvar|n}}.