Characteristic p[edit]
For a non-separable extension of characteristic p, there is nevertheless a primitive element provided the degree [E : F] is p: indeed, there can be no non-trivial intermediate subfields since their degrees would be factors of the prime p.
When [E : F] = p2, there may not be a primitive element (in which case there are infinitely many intermediate fields by Steinitz's theorem). The simplest example is , the field of rational functions in two indeterminates T and U over the finite field with p elements, and . In fact, for any in , the Frobenius endomorphism shows that the element lies in F , so α is a root of , and α cannot be a primitive element (of degree p2 over F), but instead F(α) is a non-trivial intermediate field.
![{\displaystyle f(X)=X^{p}-\alpha ^{p}\in F[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eee215dd664566a309d2efd05ca27116e6a74849)
History[edit]
In his First Memoir of 1831, published in 1846,[2] Évariste Galois sketched a proof of the classical primitive element theorem in the case of a splitting field of a polynomial over the rational numbers. The gaps in his sketch could easily be filled[3] (as remarked by the referee Poisson) by exploiting a theorem[4][5] of Lagrange from 1771, which Galois certainly knew. It is likely that Lagrange had already been aware of the primitive element theorem for splitting fields.[5] Galois then used this theorem heavily in his development of the Galois group. Since then it has been used in the development of Galois theory and the fundamental theorem of Galois theory.
The primitive element theorem was proved in its modern form by Ernst Steinitz, in an influential article on field theory in 1910, which also contains Steinitz's theorem;[6] Steinitz called the "classical" result Theorem of the primitive elements and his modern version Theorem of the intermediate fields.
Emil Artin reformulated Galois theory in the 1930s without relying on primitive elements.[7][8]