Question

Hello,

I am looking for a book that studies the set of Hurwitz quaternions (HQ). In particular, I am interested in a connection between HQ and imaginary quadratic fields (IQF); quaternion orders $\mathcal{O}(\mu)$ and ideals of the form $n[a, b + \mu]$; information on similarities and differences between ideal theory in IQF and HQ, as well as some info about the algorithms on how to perform multiplication and reduction of ideals in $\mathcal{O}(\mu)$.

Finally, I'd be glad if you recomend me some articles that study the problem of solving the equation of the form

$$ \rho\mu = \mu'\rho $$

for $\rho$. This equation allows us to "move" an ideal $n[a,b+\mu]$ with generator $\rho$ from order $\mathcal{O}(\mu)$ to order $\mathcal{O}(\mu')$. Also, if you know articles that study ambiguous ideals in HQ, I'd be glad to read them.

P.S. The only article that I am aware of is B. Venkov's "On Quaternion Arithmetic" which is pretty old (1929). I believe there exist more up-to-date articles.

Answer 1

In response to your first question (a book that studies the set of HQ), there is a book from Hurwitz, in German, Vorlesungen über die Zahlentheorie der Quaternionen, Verlag von Julius Springer, 1919, available here