A totally imaginary field is a number field with no real embeddings. A general number
field 
of degree 
has 
real embeddings 
and 
imaginary embeddings
(
),
where 
.
If 
,
is totally imaginary; if 
, it is totally real;
otherwise it has both real embeddings and imaginary
embeddings (Milne 2020, p. 90).
Totally Imaginary Field
See also
Imaginary Embedding, Imaginary Quadratic Field, Number Field Signature, Real Embedding, Totally Real FieldThis entry contributed by David Terr
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References
Milne, J. S. "Algebraic Number Theory." Version 3.08, 19 Jul 2020. https://www.jmilne.org/math/CourseNotes/ANT.pdf.Referenced on Wolfram|Alpha
Totally Imaginary FieldCite this as:
Weisstein, Eric W., with contributions by David Terr. "Totally Imaginary Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TotallyImaginaryField.html