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Apr 6, 2010 at 21:08 comment added Marty I think David's right. I guess my problem is that a closed point in Spec(P) could get sent to a nonclosed point of Spec(P_M) for all M... I'll think about it some more, but for now I humbly retract my sketchy argument.
Apr 6, 2010 at 20:39 comment added David E Speyer Something is wrong here. The only facts you are using about P is that it is the ascending union of countably many finitely generated $\mathbb{Q}$-algebras, and that has a map $\omega: P \to \mathbb{C}$ under which some elements have transcendental image. But these facts would be true of $P = \mathbb{Q}(t)$, with $\omega(t) = \pi$.
Apr 6, 2010 at 20:00 history answered Marty CC BY-SA 2.5