proof of primitive element theorem


Let F and K be arbitrary fields, and let K be an extensionPlanetmathPlanetmathPlanetmath 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.


Let F and K be fields, and let [K:F]=n be finite.

Suppose first that K=F(α). Since K/F is finite, α is algebraic over F. Let m(x) be the minimal polynomial of α over F. Now, let L be an intermediary field with FLK and let m(x) be the minimal polynomial of α over L. Also, let L be the field generated by the coefficients of the polynomial m(x). Thus, the minimal polynomial of α over L is still m(x) and LL. By the properties of the minimal polynomial, and since m(α)=0, we have a divisibility m(x)|m(x), and so:


Since we know that LL, this implies that L=L. Thus, this shows that each intermediary subfieldMathworldPlanetmath FLK corresponds with the field of definition of a (monic) factor of m(x). Since the polynomial m(x) has only finitely many monic factors, we conclude that there can be only finitely many subfields of K containing F.

Now suppose conversely that there are only finitely many such intermediary fields L. If F is a finite fieldMathworldPlanetmath, then so is K, and we have an explicit description of all such possibilities; all such extensions are generated by a single element. So assume F (and therefore K) are infiniteMathworldPlanetmathPlanetmath. Let α1,α2,,αn be a basis for K over F. Then K=F(α1,,αn). So if we can show that any field extension generated by two elements is also generated by one element, we will be done: simply apply the result to the last two elements αj-1 and αj repeatedly until only one is left.

So assume K=F(β,γ). Consider the set of elements {β+aγ} for aF×. By assumptionPlanetmathPlanetmath, this set is infinite, but there are only finitely many fields intermediate between K and F; so two values must generate the same extension L of F, say β+aγ and β+bγ. This field L contains




and so letting α=β+aγ, we see that


Title proof of primitive element theorem
Canonical name ProofOfPrimitiveElementTheorem
Date of creation 2013-03-22 14:16:27
Last modified on 2013-03-22 14:16:27
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 8
Author alozano (2414)
Entry type Proof
Classification msc 12F05