Correction Re: [math-fun] algebra question
Which should have read: < 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 . . . ") I believe that, amazingly, Gauss conjectured this (or at least had found all 9 and no others). It was proven by Heegner in the early 1950s. His proof was erroneously believed to have flaws, and it was re-proved by Baker & Stark (around 1970?). At some later point it was recognized that Heegner's proof was correct after all.
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
participants (2)
-
Bill Dubuque -
Daniel Asimov