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.