What are integers? Over $\RR$, the answer is easy, namely the infinite set \begin{equation} \ZZ = \{ …,-2,-1,0,1,2,… \} \end{equation} It is straightforward to extend this definition to the complex numbers, resulting in the Gaussian integers \begin{equation} \ZZ[\ell] = \ZZ\oplus\ZZ\ell = \{m+n\ell:m,n\in\ZZ\} \end{equation} where we continue to use $\ell$ rather than $i$ for the complex unit. The Gaussian integers form a lattice in two dimensions. The units of $\ZZ[\ell]$ are the elements with norm $1$, namely the set $\{\pm1,\pm\ell\}$.

What happens over the other division algebras?

We can of course simply extend the construction of the Gaussian integers. Over $\HH$, we obtain the Lipschitz integers \begin{equation} \ZZ[i,j,k] = \ZZ\oplus\ZZ i\oplus\ZZ j\oplus\ZZ k \end{equation} with units $\{\pm1,\pm i,\pm j,\pm k\}$. As with both the ordinary integers and the Gaussian integers, the Lipschitz integers are clearly closed under multiplication, and the norm of a Gaussian integer is an (ordinary) integer. Remarkably, the Gaussian integers are not the only subalgebra of $\HH$ with these properties.

Consider \begin{equation} q = \frac12 (1+i+j+k) \in\HH \end{equation} which has the surprising property that \begin{equation} |q| = \frac14+\frac14+\frac14+\frac14 = 1 \in\ZZ \end{equation} so that the norm of $q$ is an integer, even though the components of $q$ are not. Such “half-integer” quaternions do not quite close under multiplication, since for instance \begin{equation} \left(\frac12 (1+i+j+k)\right) \left(\frac12 (1-i+j+k)\right) = k \end{equation} which has integer coefficients rather than half-integer. But the union of the “half-integer” quaternions with the “integer” quaternions, that is, with the Lipschitz integers, does close under multiplication. The elements of this union are called Hurwitz integral quaternions, or simply Hurwitz integers, and can be generated by $\{q,i,j,k\}$, that is, any Hurwitz integer can be written as a product of these four generators. The units among the Hurwitz integers are $\{\pm1,\pm i,\pm j,\pm k,\frac12(\pm1\pm i\pm j\pm k)\}$, where all possible combinations of signs are permitted in the last element. As the subalgebra consisting of all rational quaternions with integer norm, there are contexts in which the Hurwitz integers, rather than the Lipschitz integers, play the role of “quaternionic integers”.

Why did we add the qualifier “rational”? There are of course other elements of $\HH$ and $\CC$ with unit norm. For instance, the cube roots of unity over $\CC$, which are $\{1,\frac12(-1\pm\sqrt3\ell)\}$, not only have unit norm but form a subalgebra of $\CC$, that is, they close under multiplication. We could therefore have considered the Eisenstein integers $\{m+n\frac12(-1+\sqrt3\ell)\}$, which are also called Euler integers. However, the unit elements $\pm\ell$ are not Eisenstein integers—hence the restriction to rational coefficients.

A remarkable property of the Hurwitz integers is that they contain rational cube (and sixth) roots of unity. Because \begin{equation} q^3 = -1 \end{equation} the Hurwitz integer \begin{equation} -q^2 = \frac12 (1-k-j-k) \end{equation} is a rational cube root of unity.

Further discussion of integral quaternions can be found in [Conway and Smith] .