Primitive Element Theorem - Existence Statement

Existence Statement

The interpretation of the theorem changed with the formulation of the theory of Emil Artin, around 1930. From the time of Galois, the role of primitive elements had been to represent a splitting field as generated by a single element. This (arbitrary) choice of such an element was bypassed in Artin's treatment. At the same time, considerations of construction of such an element receded: the theorem becomes an existence theorem.

The following theorem of Artin then takes the place of the classical primitive element theorem.

Theorem

Let be a finite degree field extension. Then for some element if and only if there exist only finitely many intermediate fields K with .

A corollary to the theorem is then the primitive element theorem in the more traditional sense (where separability was usually tacitly assumed):

Corollary

Let be a finite degree separable extension. Then for some .

The corollary applies to algebraic number fields, i.e. finite extensions of the rational numbers Q, since Q has characteristic 0 and therefore every extension over Q is separable.