Questions tagged [totally-real-field]

For questions about number fields that are totally real, i.e. whose every complex embedding has real image.

A number field is totally real if the image every complex embedding is contained in the reals. Equivalently, if it is generated over the rationals by one root of a polynomial with only real roots. Totally real fields are closely related to CM-fields, which are (non-real) quadratic extensions of totally real fields.

6 questions
1
vote
0 answers

Example of a totally real number that is neither totally positive nor totally negative?

Say that an algebraic number $\alpha\in \mathbb{C}$ is totally real iff $\mathbb{Q}(\alpha)$ is a totally real number field. Why does the set of all totally real numbers form a subfield of $\mathbb{R}$? What is an example of a totally real number…