primitive element theorem

Theorem 1.

Let F and K be arbitrary fields, and let K be an extensionPlanetmathPlanetmath of F of finite degree. Then there exists an element αK such that K=F(α) if and only if there are finitely many fields L with FLK.

Note that this implies that every finite separable extensionMathworldPlanetmath is not only finitely generatedMathworldPlanetmathPlanetmath, it is generated by a single element.

Let X be an indeterminateMathworldPlanetmath. Then (X,i) is not generated over by a single element (and there are infinitely many intermediate fields (X,i)/L/). To see this, suppose it is generated by an element α. Then clearly α must be transcendental, or it would generate an extension of finite degree. But if α is transcendental, we know it is isomorphicPlanetmathPlanetmathPlanetmath to (X), and this field is not isomorphic to (X,i): for example, the polynomialMathworldPlanetmathPlanetmathPlanetmath Y2+1 has no roots in the first but it has two roots in the second. It is also clear that it is not sufficient for every element of K to be algebraic over F: we know that the algebraic closureMathworldPlanetmath of has infinite degree over , but if α is algebraic over then [(α):] will be finite.

This theorem has the corollary:

Corollary 1.

Let F be a field, and let [F(β,γ):F] be finite and separable. Then there exists αF(β,γ) such that F(β,γ)=F(α). In fact, we can always take α to be an F-linear combinationMathworldPlanetmath ( of β and γ.

To see this (in the case of characteristicPlanetmathPlanetmath 0), we need only show that there are finitely many intermediate fields. But any intermediate field is contained in the splitting fieldMathworldPlanetmath of the minimal polynomialsPlanetmathPlanetmath of β and γ, which is Galois with finite Galois groupMathworldPlanetmath. The explicit form of α comes from the proof of the theorem.

For more detail on this theorem and its proof see, for example, Field and Galois TheoryMathworldPlanetmath, by Patrick Morandi (Springer Graduate Texts in Mathematics 167, 1996).

Title primitive element theorem
Canonical name PrimitiveElementTheorem
Date of creation 2013-03-22 11:45:48
Last modified on 2013-03-22 11:45:48
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 18
Author alozano (2414)
Entry type Theorem
Classification msc 12F05
Classification msc 65-01
Related topic SimpleFieldExtension
Related topic PrimitiveElementOfBiquadraticField2
Related topic PrimitiveElementOfBiquadraticField