Show that primes of the form are of the form .
(Note : since primes of the form are of the form , the difficulty of Lagrange's four square theorem is thus reduced to primes of the form .)
Alright, so nobody has answered this problem so I'll give my answer.
For , both are quadratic residues; therefore is a quadratic residue.
For , neither are quadratic residues; therefore is a quadratic residue.
Therefore we can find such that , i.e. . Since is a unique factorization domain, and since neither factor on the left is divisible by , we deduce that is not prime in , and admits a factorization . Taking norms we get , so .
Let us prove this congruence inequality. Define and consider for . There are such choices for and , therefore, for by pigeonholing modulo . Thus, . Set . Thus, we get that . It must be the case that because otherwise this would force which would be a contradiction. Thus, but they also satisfy by construction. Thus, we have found a solution for which satisfies this inequality.
If then and if then . Thus, in either case there exists such that . By above there exists a solution to with . Then, . Thus, . This means, by definition, there exists such that . However, . This forces, . If then and the proof is complete. Otherwise, . But then this forces to be even, so . This implies . In either case it is always possible to express like so.
It follows from the fact that at most two prime ideals can lie over a prime in a quadratic extension. Since are two ideals lying over , there can be no other such ideals. This implies that there can be no other element of norm : any other such element would generate another principal ideal of norm , which is impossible.
I asked the question about uniqueness of expression because I seen lots of people simply say, "uniquness in factorization implies uniquness of expression" but a lot of times this kind of argument is not so easy. I have seen expression problems which do not even have a uniquness up to , they were more complicated. Usually, if there are are many units then the express is "less unique".