Note that Z[i] is called the "Gaussian integers" -- you don't have to keep repeating "the ring of integers with i adjoined"! Also I should point out that in ge

Note that Z[i] is called the "Gaussian integers" -- you don't have to keep repeating "the ring of integers with i adjoined"! Also I should point out that in general Z[r] for some r (note obviously r doesn't have to be real!) will contain more than just a+br; it'd only be just a+br if r satisfies a monic quadratic over Z. (I also have to nitpick and point out that these results apply to rings of integers -- actually more broadly -- but not to the fields. Yeah unfortunately mathematicians often abuse the language here, using "number field" to refer to the ring, and it's annoying. But Hilbert's 10th for Q remains open to my knowledge.) Edit: Ugh I forgot this site doesn't allow bold