Primitive element (field theory) Guide, Meaning , Facts, Information and Description

In mathematics, a primitive element for an extension of fieldss 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 implies that L is either a finite extension of K, in case ζ is an algebraic element of L over K, or that 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, almost fully, 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

X² − 2

and

X² − 3,

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 γⁱ 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.

In general, any field extension L/K of finite degree is evidently generated by some finite set of elements. By induction it would be enough to show how to write K(α, β) as K(γ) for some suitable γ. However, this is not possible in full generality - it is not true unless the extension is assumed to be a separable extension.

The primitive element theorem therefore states that a finite separable field extension has a primitive element. The proof proceeds by induction, as suggested: assuming that K is also an infinite field, some K-linear combination of α and β can be chosen as γ. When, on the other hand, K is finite, the structure of finite fields can be applied to give a proof.

When the extension is allowed to be inseparable, 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] = p² but no elements of L having degree p² over K, as a primitive element must have.

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