Totally real number field

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.

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.

The number field Q (√2) sits inside R , and the two embeddings of the field into C send every element in the field to another element of R , hence the field is totally real.