Let A be an étale algebra over Q with components K1 x ... x Kn. We say that A is a CM-étale algebra if every component Ki is a CM-field, that is, Ki has an involution that acts as applying complex conjugation after applying any homomorphism to the complex numbers. If A is a CM-étale algebra, then it has an involution with the same property. For this reason, we call this involution complex conjugation and denote it as /line(.).
Given an element of A, an order or a fractional ideal in A, we say that it is conjugate stable if it equals its complex conjugate. An element x of A is called totally real if x=/line(x) and totally imaginary if x= - /line(x). A totally real element a is called totally positive (resp. totally negative) if varphi(a) > 0 (resp. varphi(a)<0) for every homomorphism varphi: A to mathbb(C) .
Returns if the algebra A is the product of CM fields.
If the algebra A of the element x is a product of CM fields, it returns the complex conjugate of the argument.
Given an order O in a CM-étale algebra, it returns whether O is conjugate stable and the complex conjugate.
Given an order O in a CM-étale algebra, it returns the complex conjugate of O.
Given a fractional ideal I in a CM-étale algebra, it returns whether I is conjugate stable and the complex conjugate. Note: if the order of I is not conjugate stable, then the second output will be defined over the complex conjugate of the order.
If A is a product of CM fields, it returns the complex conjugate of the fractional ideal I. Note: if the order of I is not conjugate stable, then the output will be defined over the complex conjugate of the order.