Primitive Element Theorem - Counterexamples

Counterexamples

For non-separable extensions, necessarily in characteristic p with p a prime number, then at least when the degree is p, L / K has a primitive element, because there are no intermediate subfields. When = p2, there may not be a primitive element (and therefore there are infinitely many intermediate fields). This happens, 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. In fact one can see that for any α in L, the element αp lies in K, but a primitive element must have degree p2 over K.