Totally real number field
- 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 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 algebra 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 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.
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