Properties
- If is an algebraic field extension, and if are separable over F, then and are separable over F. In particular, the set of all elements in E separable over F forms a field.
- If is such that and are separable extensions, then is separable. Conversely, if is a separable algebraic extension, and if L is any intermediate field, then and are separable extensions.
- If is a finite degree separable extension, then it has a primitive element; i.e., there exists with . This fact is also known as the primitive element theorem or Artin's theorem on primitive elements.
Read more about this topic: Separable Extension
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)