existence of the minimal polynomial

Proposition 1.

Let K/L be a finite extensionMathworldPlanetmath of fields and let kK. There exists a unique polynomialMathworldPlanetmathPlanetmathPlanetmath mk(x)L[x] such that:

  1. 1.

    mk(x) is a monic polynomialMathworldPlanetmath;

  2. 2.


  3. 3.

    If p(x)L[x] is another polynomial such that p(k)=0, then mk(x) divides p(x).


We start by defining the following map:


Note that this map is clearly a ring homomorphismMathworldPlanetmath. For all p(x),q(x)L[x]:

  • ψ(p(x)+q(x))=p(k)+q(k)=ψ(p(x))+ψ(q(x))

  • ψ(p(x)q(x))=p(k)q(k)=ψ(p(x))ψ(q(x))

Thus, the kernel of ψ is an ideal of L[x]:


Note that the kernel is a non-zero ideal. This fact relies on the fact that K/L is a finite extension of fields, and therefore it is an algebraic extensionMathworldPlanetmath, so every element of K is a root of a non-zero polynomial p(x) with coefficients in L, this is, p(x)Ker(ψ).

Moreover, the ring of polynomials L[x] is a principal ideal domainMathworldPlanetmath (see example of PID). Therefore, the kernel of ψ is a principal idealMathworldPlanetmath, generated by some polynomial m(x):


Note that the only units in L[x] are the constant polynomials, hence if m(x) is another generator of Ker(ψ) then


Let α be the leading coefficient of m(x). We define mk(x)=α-1m(x), so that the leading coefficient of mk is 1. Also note that by the previous remark, mk is the unique generator of Ker(ψ) which is monic.

By construction, mk(k)=0, since mk belongs to the kernel of ψ, so it satisfies (2).

Finally, if p(x) is any polynomial such that p(k)=0, then p(x)Ker(ψ). Since mk generates this ideal, we know that mk must divide p(x) (this is property (3)).

For the uniqueness, note that any polynomial satisfying (2) and (3) must be a generator of Ker(ψ), and, as we pointed out, there is a unique monic generator, namely mk(x).

Title existence of the minimal polynomial
Canonical name ExistenceOfTheMinimalPolynomial
Date of creation 2013-03-22 13:57:24
Last modified on 2013-03-22 13:57:24
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 7
Author alozano (2414)
Entry type Theorem
Classification msc 12F05
Related topic FiniteExtension
Related topic AlgebraicMathworldPlanetmath