Some primes in the ring of integers (17, for example) cease to behave as such in the ring of gaussian integers, while others (7, for instance) keep being prime there as well. The former are of the form $4n +1$, while the later can be written as $4n + 3$.
I would like to know who was responsible for this finding? Any reference? Was it Kummer? Or was it already Gauss? Quotation, bibliography? Thanks in advance.
Gauss introduced Gaussian integers and proved unique factorization in the ring in the same paper.
The theorem about integer primes that can be written as the sum of squares was already well-known, having been first proved by Euler.
I have no doubt that Gauss was aware that the integer primes that factor in $\mathbb Z[i]$ were exactly those of the form $4n+1$ (and, of course, $2$.)