Totally real number field

In number theory, a number field K is called totally real if for each embedding of K into the complex numbers the image (mathematics) lies inside the real numbers. Equivalent conditions on K , a finite extension of the rational number field Q , are that K is generated over Q by one root of an integer polynomial P , all of the roots of P being real; or that the tensor product of fields of K with the real field, over Q , is a product of copies of R .

For example, quadratic fields K of degree 2 over Q are either real (and then totally real), or complex, depending on whether the square root of a positive or negative number is adjoined to Q . In the case of cubic fields, a cubic integer polynomial P irreducible polynomial over Q will have at least one real root. If it has one real and two complex roots the corresponding cubic extension of Q defined by adjoining the real root will not be totally real, although it is a field of real numbers.

The totally real number fields play a significant special role in algebraic number theory. An abelian extension of Q is either totally real, or contains a totally real subfield over which it has degree two.