3 ms·
Something that may not be clear when reading this is the distinction between complex numbers and the ring of integers adjoined with i. "Complex numbers" are of
by empath75 2y ago
Something that may not be clear when reading this is the distinction between complex numbers and the ring of integers adjoined with i.
"Complex numbers" are of the form a+bi where a and b can be any real number -- 1, 5, pi, the square root of 2, -2.14, etc.
The ring of integers adjoined with i are numbers of the form a+bi where a and b are both integers (-1, 5, 34, etc).
You can also, in addition to using i, adjoin any real number to the integers and get a new field with numbers of the form a+bx where a and b are integers and x is any additional number you want to add-- frequently square roots like the square root of two.
This result shows undecidability of diophantine equations in all those fields of integers, but not complex numbers, for which it's easy to prove that there are _always_ solutions.
- Sniffnoy 2y agoNote 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
- deleted 2y ago[deleted]