Primitive element theorem

The set of primitive element is a mathematical theorem from algebra that specifies sufficient conditions that a field extension is a simple field extension. Are body, then the body extension is simply called, if it can be generated by adjoining a single element. Such will generally not be uniquely determined element is called a primitive element. The set of primitive element has been fully proved by Galois and can be found in a publication by Abel from 1829.

Set

There are two sets, referred to as a set of primitive element, wherein the second set is a consequence of the first.

• A field extension is easy if is of the form with an algebraic element over and over separable elements.
• Every finite, separable field extension is simple.

Importance

In particular, finite Galois extensions of this form and therefore easy. If such an extension, so is an element of the Galois group, that is a - automorphism of, already uniquely determined by the value. Therefore, the meaning of this sentence results in Galois theory.