If I understand correctly, any quaternion algebra over the rationals is a noncommutative associative division algebra. I am currently working with implementations of quaternion algebras in MAGMA and have encountered the following situation:
If I generate a quaternion algebra D(i,j,k) := QuaternionAlgebra( Rationals()| 1, -2) and try to find an inverse to the element j+k, (j+k)$^{-1}$, MAGMA returns that it cannot take negative powers of non-units. In fact, we have that (j+k)$^2$ = 0 which should not be possible in an associative division algebra. Can anyone tell me what my mistake is?
Shouldn't all quaternion algebras generated by non-zero rational numbers a, b such that i$^2$ = a, j$^2$ = b be skew fields?