There is a very nice and readable little book by Hurwitz from 1909a very nice and readable little book by Hurwitz from 1919 (Adolf Hurwitz, Vorlesungen über die Zahlentheorie der Quaternionen, Springer Verlag 1919) in which he describes the properties of the integer quaternions (later called Hurwitzian quaternions): the subring of the Hamiltonian quaternions of the form a + bi + cj + dk where a, b, c, and d are either all integers or all half-integers (like 3/2). Basically the point of the book is that this ring behaves pretty much like any other UFD, except that it is not commutative. So there are prime elements and unique factorization into them etc, when using the right notion of uniqueness to take care of the left/rigth business. Towards the end of the book he starts considering the finite quotients obtained by modding out the two sided ideals generated by these primes and doing a lot of counting inside these rings he deduces the four square theorem. It is not the most beautiful part of the book because there is a lot of 'administration' to keep track of, but it is a nice elementary proof of the theorem. I hope I can find the title.
There is a very nice and readable little book by Hurwitz from 1909 in which he describes the properties of the integer quaternions (later called Hurwitzian quaternions): the subring of the Hamiltonian quaternions of the form a + bi + cj + dk where a, b, c, and d are either all integers or all half-integers (like 3/2). Basically the point of the book is that this ring behaves pretty much like any other UFD, except that it is not commutative. So there are prime elements and unique factorization into them etc, when using the right notion of uniqueness to take care of the left/rigth business. Towards the end of the book he starts considering the finite quotients obtained by modding out the two sided ideals generated by these primes and doing a lot of counting inside these rings he deduces the four square theorem. It is not the most beautiful part of the book because there is a lot of 'administration' to keep track of, but it is a nice elementary proof of the theorem. I hope I can find the title.
There is a very nice and readable little book by Hurwitz from 1919 (Adolf Hurwitz, Vorlesungen über die Zahlentheorie der Quaternionen, Springer Verlag 1919) in which he describes the properties of the integer quaternions (later called Hurwitzian quaternions): the subring of the Hamiltonian quaternions of the form a + bi + cj + dk where a, b, c, and d are either all integers or all half-integers (like 3/2). Basically the point of the book is that this ring behaves pretty much like any other UFD, except that it is not commutative. So there are prime elements and unique factorization into them etc, when using the right notion of uniqueness to take care of the left/rigth business. Towards the end of the book he starts considering the finite quotients obtained by modding out the two sided ideals generated by these primes and doing a lot of counting inside these rings he deduces the four square theorem. It is not the most beautiful part of the book because there is a lot of 'administration' to keep track of, but it is a nice elementary proof of the theorem. I hope I can find the title.
There is a very nice and readable little book by Hurwitz from 1909 in which he describes the properties of the integer quaternions (later called Hurwitzian quaternions): the subring of the Hamiltonian quaternions of the form a + bi + cj + dk where a, b, c, and d are either all integers or all half-integers (like 3/2). Basically the point of the book is that this ring behaves pretty much like any other UFD, except that it is not commutative. So there are prime elements and unique factorization into them etc, when using the right notion of uniqueness to take care of the left/rigth business. Towards the end of the book he starts considering the finite quotients obtained by modding out the two sided ideals generated by these primes and doing a lot of counting inside these rings he deduces the four square theorem. It is not the most beautiful part of the book because there is a lot of 'administration' to keep track of, but it is a nice elementary proof of the theorem. I hope I can find the title.