6
$\begingroup$

Is there a commutative integral domain $R$ in which:

  • every nonzero prime ideal $Q$ is maximal, and
  • for every prime power $q\equiv 3 \bmod 8$, there is a maximal ideal $Q$ of $R$ such that $R/Q$ is a field of size $q$?

I am looking for a ring where "$q\equiv 3 \bmod 8$" describes finite fields, rather than just finite local rings like in the ring $\mathbb{Z}$. The only examples I can think of that satisfy the first condition have a finite number of residue fields of each characteristic. The only examples I can think of that satisfy the second condition have a ton of non-maximal non-proper prime ideals with all sorts of bizarre fields associated to them.

  • 0
    To be honest, I just assumed R was some silly universal ring covered in every algebraic number theory class and that people would just say "oh you mean the completion of the adele ring" or "oh you want the ring of all algebraic integers" or something. If the ring is too horrible or unknown, then I guess it isn't too useful to me.2010-12-20

0 Answers 0