13 Sep
2006
13 Sep
'06
5:59 p.m.
Daniel Asimov <dasimov@earthlink.net> wrote:
Bill Cordwell writes:
According to Grove, there are nine rings of algebraic integers over Q[sqrt(m)] that are PIDs, where m is a negative integer, viz., m = -1,-2,-3,-7,-11,-19,-43,-67, and -163. These are then UFDs, but are they the only such UFDs?
Yes, the only UFD's of the form Z[sqrt(-n)] for a positive integer n. -- (NOT "the only PID's . . . ")
They are equivalent: PID <-> UFD of dimension <= 1 i.e. except for degenerate field case (dimension = 0), a domain is a PID iff it is a UFD with maximal primes (the proof is easy). Since number / Dedekind domains are of dimension <= 1 it follows they are PID <-> UFD --Bill Dubuque