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?
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[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? 

