### Creating maximal orders in quaternion algebras

Posted:

**Wed May 16, 2018 11:49 am**I am currently trying to implement the following setting in MAGMA:

Let D be a skew field that is central and finite-dimensional over a number field F (more specifically: a quaternion algebra). Let Delta (subset of D) be a maximal order over the ring of integers of F ("OF").

I have tried implementing several versions of this setting, sometimes using

the rationals as RationalsAsNumberField() (since the documentation for Quaternion Algebras says that some MAGMA intrinsics are only implemented for number fields), sometimes with F as a quadratic field, but none of them worked so far. In particular, I wasn't able to create the maximal OF-order Delta.

My MAGMA code looks like this:

The error I get when executing this code is:

Let D be a skew field that is central and finite-dimensional over a number field F (more specifically: a quaternion algebra). Let Delta (subset of D) be a maximal order over the ring of integers of F ("OF").

I have tried implementing several versions of this setting, sometimes using

the rationals as RationalsAsNumberField() (since the documentation for Quaternion Algebras says that some MAGMA intrinsics are only implemented for number fields), sometimes with F as a quadratic field, but none of them worked so far. In particular, I wasn't able to create the maximal OF-order Delta.

My MAGMA code looks like this:

(I am aware that this order is not maximal.)> F6:=QuadraticField(2);

> OF6:=RingOfIntegers(F6);

> D6<i,j,k>:=QuaternionAlgebra<F6|-1,-1>;

> Seq6:=[1,i,j,k];

> Delta6:=Order(OF6,Seq6);

The error I get when executing this code is:

Can anybody help me fix this problem?> Delta6:=Order(OF6,Seq6);

>Runtime error in 'Order': Coefficient ring must be a

polynomial ring or integer ring