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Concept
Totally real number field
Summary
In number theory, a number field F is called totally real if for each embedding of F into the complex numbers the lies inside the real numbers. Equivalent conditions are that F 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 F with the real field, over Q, is isomorphic to a tensor power of R.
For example, quadratic fields F 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.
Any number field that is Galois over the rationals must be either totally real or totally imaginary.
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