Let L be a finitely generated
separable extension of a field K of characteristic p≠0. Artin’s theorem of a primitive
element states that if L is algebraic over K, then L is a simple extension of K. If L is
non-algebraic over K, then an element 𝜃 ∈ L with the property L = L′(𝜃) for every
L′, L ⊇ L′⊇ K, such that L is separable algebraic over L′ is called a generalized
primitive element for L over K. The main result states that if [K : Kp] > p, then
there exists a generalized primitive element for L over K. An example is given
showing that if [K : Kρ] ≦ p, then L need not have a generalized primitive element
over K.
