# Predicates on Algebras

A quaternion algebra $A$ over a number field $F$ with $[F:{\mathbb{Q}}]=h$ is *definite* (or *totally definite*) if $F$ is totally real and $A \otimes_{{\mathbb{Q}}} {\mathbb{R}}\cong H^h$, where $H$ is the division ring of real Hamiltonians, otherwise $A$ is *indefinite*.

A quaternion algebra $A$ over ${\mathbb{F}}_q(X)$ is called *definite* if the place corresponding to the degree valuation is ramified.

## `IsDefinite(A): AlgQuat -> BoolElt`

## `IsIndefinite(A): AlgQuat -> BoolElt`

Given a quaternion algebra $A$ over a number field, ${\mathbb{Q}}$ or ${\mathbb{F}}_q(X)$ with $q$ odd, returns `true` if and only if $A$ is a (totally) definite or indefinite quaternion algebra, respectively.