§ Primitive element theorem

§ Try 0: Naive attempt

§ Part 1: Primitive element iff number of intermediate subfields is finite

§ Forward: Finitely many intermediate subfields implies primitive

§ Backward: primitive implies finitely many intermediate subfields

§ Interlude: finite extension with infinitely many subfields

§ Part 2: If E/kE/k is finite and separable then it has a primitive element

§ References