# Primitive element field theory

In mathematics, a **primitive element** for an extension of fields *L*/*K* is an element ζ of *L* such that

*L*=*K*(ζ),

or in other words such that *L* is generated by ζ over *K*. This means that every element of *L* can be written as a quotient of two polynomials in ζ with coefficients from *K*.

If the extension *L*/*K* admits a primitive element, then *L* is either a finite extension of *K*, in case ζ is an algebraic element of *L* over *K*, or *L* is isomorphic to the field of rational functions over *K* in one indeterminate, if ζ is a transcendental element of *L* over *K*.

The **primitive element theorem** of field theory answers the question of which finite field extensions have primitive elements. It is not, for example, immediately obvious that if one adjoins to the field **Q** of rational numbers roots of both polynomials

*X*− 2^{2}

and

*X*− 3,^{2}

say α and β respectively, to get a field *K* = **Q**(α, β) of degree 4 over **Q**, that *K* is **Q**(γ) for a primitive element γ. One can in fact check that with

- γ = α + β

the powers γ^{i} for 0 ≤ *i* ≤ 3 can be written out as linear combinations of 1, α, β and αβ with integer coefficients. Taking these as a system of linear equations, one can solve for α and β over **Q**, which implies that this choice of γ is indeed a primitive element in this example.

The general **primitive element theorem** states:

- The field extension
*L*/*K*is finite and has a primitive element if and only if there are only finitely many intermediate fields*F*with*K*⊆*F*⊆*L*.

In this form, the theorem is somewhat unwieldy and rarely used. An important corollary states

- Every finite separable extension
*L*/*K*has a primitive element.

This corollary applies to the example considered above (and to many others like it), since **Q** has characteristic 0 and therefore every extension over **Q** is separable.

For non-separable extensions, one can at least state the following:

- If the degree [
*L*:*K*] is a prime number, then*L*/*K*has a primitive element.

If the degree is not a prime number and the extension is not separable, one can give counterexamples. For example if *K* is *F _{p}*(

*T*,

*U*), the field of rational functions in two indeterminates

*T*and

*U*over the finite field with

*p*elements, and

*L*is obtained from

*K*by adjoining a

*p*-th root of

*T*, and of

*U*, then there is no primitive element for

*L*over

*K*. In fact one can see that for any α in

*L*, the element α

^{p}lies in

*K*. Therefore we have [

*L*:

*K*] =

*p*

^{2}but there is no elements of

*L*with degree

*p*

^{2}over

*K*, as a primitive element must have.