algebraic number field
a finite degree (and hence algebraic) field extension of the field of rational numbers
quadratic field
algebraic number field of degree two over the rational numbers
totally imaginary number field
a number field that cannot be embedded in the real numbers
cyclotomic field
field extension of the rational numbers by a primitive root of unity
totally real number field
a number field K such that, for each embedding of K into the complex numbers, the image lies inside the real numbers