Re: [math-fun] algebra question

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?

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Yes, the only PID's of the form Z[sqrt(-n)] for a positive integer n. 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. --Dan