Algebraic extension

In a body of the ring extension is algebraically, if each element of algebraic is, i.e., if each element of the zero of a polynomial with coefficients to be. Body extensions that are not algebraic, ie contain transcendental elements are called transcendental.

For example, the extensions and algebraic, while transcendent.

If an upper body then may be considered as vector space and determine its dimension. This vector space dimension is called the degree of the field extension. Depending on whether this degree is finite or infinite, is also called the field extension is finite or infinite. Each transcendental extension is infinite. It follows that any finite algebraic extension.

But there are also infinite algebraic extensions, for example, form the algebraic numbers an infinite extension of

Is algebraic over then is the ring of all polynomials in more than even a body. is a finite algebraic extension of such extensions which arise by adjoining a single element, are called simple extensions.

A body which has no real algebraic extension is algebraically closed.

Are and field extensions, the following statements are equivalent:

  • Is algebraic.
  • And are algebraically.