Editing Hurwitz quaternion (section)

==Factorization into irreducible elements==

A Hurwitz integer is called irreducible if it is not 0 or a [[unit (ring theory)|unit]] and is not a product of non-units. 
A Hurwitz integer is irreducible [[if and only if]] its norm is a [[prime number]]. The irreducible quaternions are sometimes called prime quaternions, but this can be misleading as they are not [[prime element|primes]] in the usual sense of [[commutative algebra]]: it is possible for an irreducible quaternion to divide a product ''ab'' without dividing either ''a'' or ''b''. Every Hurwitz quaternion can be factored as a product of irreducible quaternions. This factorization is not in general unique, even up to units and order, because a positive odd prime ''p'' can be written in 24(''p''+1)  ways as a product of two irreducible Hurwitz quaternions of norm ''p'', and for large ''p'' these cannot all be equivalent under left and right multiplication by units as there are only 24 units. However, if one excludes this case then there is a version of unique factorization. More precisely, every Hurwitz quaternion can be written uniquely as the product of a positive integer and a primitive quaternion (a Hurwitz quaternion not divisible by any integer greater than 1). The factorization of a primitive quaternion into irreducibles is unique up to order and units in the following sense: if 
:''p''<sub>0</sub>''p''<sub>1</sub>...''p''<sub>''n''</sub>
and 
:''q''<sub>0</sub>''q''<sub>1</sub>...''q''<sub>''n''</sub>
are two factorizations of some primitive Hurwitz quaternion into irreducible quaternions where ''p''<sub>''k''</sub> has the same norm as ''q''<sub>''k''</sub> for all ''k'', then

: <math>\begin{align}
q_0 & = p_0 u_1 \\
q_1 & = u_1^{-1} p_1 u_2 \\
& \,\,\,\vdots \\
q_n & = u_n^{-1} p_n
\end{align}</math>
for some units ''u''<sub>''k''</sub>.