It is wellknown that there exists a (justly celebrated) topological proof of the infinitude of primes (Hillel Fürstenburg, 1955). Does there also exist an algebraic proof?
closed as too localized by Andy Putman, Felipe Voloch, Gerry Myerson, Yemon Choi, Cam McLeman Oct 5 '10 at 4:25This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question. 


[I don't really know what constitutes an "algebraic proof" of infinitude of prime numbers.] The proof that BCnrd alludes to above is described in somewhat more length on p. 5 of http://math.uga.edu/~pete/4400primes.pdf For those who have seen this argument before: I would like to actually include a citation to something written by Washington but I have not been able to find such a document. Does anyone know of one? It is also possible to prove more general algebraic results by Euclidstyle proofs. One such result appeared on a UGA qualifying exam in algebra some years ago:
As Mr. Bill Dubuque pointed out on another forum, this problem goes back at least as far as Kaplansky's Commutative Rings book. He also remarked that it is no harder to prove a slight generalization: if $R$ is ininite and $\# R > \# R^{\times})$, then $R$ has infinitely many maximal ideals. I also posted the following question on the other forum several years ago: what is an example of a ring satisfying the hypotheses of this result for which it would otherwise be difficult to see that it has infinitely many maximal ideals? 

