splitting field of a finite set of polynomials

Lemma 1.

(Cauchy,Kronecker) Let K be a field. For any irreducible polynomialMathworldPlanetmath f in K[X] there is an extension fieldMathworldPlanetmath of K in which f has a root.


If I is the ideal generated by f in K[X], since f is irreduciblePlanetmathPlanetmath, I is a maximal idealMathworldPlanetmath of K[X], and consequently K[X]/I is a field.
We can construct a canonical monomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath v from K to K[X]. By tracking back the field operation on K[X]/I, v can be extended to an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath w from an extension field L of K to K[X]/I.
We show that α=w-1(X+I) is a root of f.
If we write f=i=1nfiXi then f+I=0 implies:

w(f(α)) =w(i=1nfiαi)

which means that f(α)=0.∎

Theorem 1.

Let K be a field and let M be a finite set of nonconstant polynomialsMathworldPlanetmathPlanetmathPlanetmath in K[X]. Then there exists an extension field L of K such that every polynomial in M splits in L[X]


If L is a field extension of K then the nonconstant polynomials f1,f2,,fn split in L[X] iff the polynomial i=1nfi splits in L[X]. Now the proof easily follows from the above lemma. ∎

Title splitting fieldMathworldPlanetmath of a finite set of polynomials
Canonical name SplittingFieldOfAFiniteSetOfPolynomials
Date of creation 2013-03-22 16:53:09
Last modified on 2013-03-22 16:53:09
Owner polarbear (3475)
Last modified by polarbear (3475)
Numerical id 16
Author polarbear (3475)
Entry type Theorem
Classification msc 12F05