# Totally Real and Positive

## `IsTotallyReal(a): AlgEtQElt -> BoolElt`

Returns whether element $a$ is totally real.

## `IsTotallyRealPositive(a): AlgEtQElt -> BoolElt`

Returns whether the element $a$ is totally positive, that is, totally real and with positive image in CC.

## `TotallyRealSubAlgebra(K): AlgEtQ -> AlgEtQ, Map`

Given a CM algebra $K$, returns the unique totally real subalgebra, with an embedding.

## `TotallyRealUnitGroup(S): AlgEtQOrd -> Grp`

Given an order $S$ in a CM étale algebra $A$ returns the groups of totally real units of $S$, as a subgroup of $S^*$.

## `TotallyRealPositiveUnitGroup(S): AlgEtQOrd -> Grp`

Given an order $S$ in a CM étale algebra. Returns the groups of totally positive units of $S$, as a subgroup of $S^*$.