totally real and imaginary fields

For this entry, we follow the notation of the entry real and complex embeddings.

Let K be a subfieldMathworldPlanetmath of the complex numbersMathworldPlanetmathPlanetmath, , and let ΣK be the set of all embeddings of K in .

Definition 1.

With K as above:

  1. 1.

    K is a totally real field if all embeddings ψΣK are real embeddings.

  2. 2.

    K is a totally imaginary field if all embeddings ψΣK are (non-real) complex embeddings.

  3. 3.

    K is a CM-field or complex multiplicationMathworldPlanetmath field if K is a totally imaginary quadratic extension of a totally real field.

Note that, for example, one can obtain a CM-field K from a totally real number field F by adjoining the square root of a number all of whose conjugatesPlanetmathPlanetmath are negative.

Note: A complex number ω is real if and only if ω¯, the complex conjugateMathworldPlanetmath of ω, equals ω:


Thus, a field K which is fixed pointwise by complex conjugation is real (i.e. strictly contained in ). However, K might not be totally real. For example, let α be the unique real third root of 2. Then (α) is real but not totally real.

Given a field L, the subfield of L fixed pointwise by complex conjugation is called the maximal real subfield of L.

For examples (of (1),(2) and (3)), see examples of totally real fields.

Title totally real and imaginary fields
Canonical name TotallyRealAndImaginaryFields
Date of creation 2013-03-22 13:55:02
Last modified on 2013-03-22 13:55:02
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 8
Author alozano (2414)
Entry type Definition
Classification msc 12D99
Synonym complex multiplication field
Related topic RealAndComplexEmbeddings
Related topic TotallyImaginaryExamplesOfTotallyReal
Related topic ExamplesOfRamificationOfArchimedeanPlaces
Defines totally real field
Defines totally imaginary field
Defines CM-field
Defines maximal real subfield