Primitive element theorem

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In mathematics, more specifically in field theory, the primitive element theorem provides a characterization of the finite field extensions which are simple and thus can be generated by the adjunction of a single primitive element.

[edit] Primitive element theorem

A 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.

In more concrete language, every separable extension L/K of finite degree n is generated by a single element x satisfying a polynomial equation of degree n, xⁿ +c₁x^n-1+..+c_n=0, with coefficients in K. The primitive element x provides a basis [1,x,x²,...,x^n-1] for L over K.

This corollary applies to algebraic number fields, which are finite extensions of the rational numbers Q, 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] = p² but there is no element of L with degree p² over K, as a primitive element must have.

[edit] Example

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 the extension is simple and there exists a primitive element γ in K so that K = Q(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, or by factoring, one can solve for α and β over Q(γ), which implies that this choice of γ is indeed a primitive element in this example.

More generally, the set of all primitive elements for a finite separable extension L/K is the complement of a finite collection of proper subspaces of L.