[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 Euclid-style proofs. One such result appeared on a UGA qualifying exam in algebra some years ago: first Prof. D. Lorenzini mentioned that he had seen the problem it to me and remarked that it gave had given him some pause.
Show that an infinite commutative ring $R$ with finite unit group $R^{\times}$ has infinitely many maximal ideals.
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 $# \# R > > \max(\aleph_0, \# R^{\times})$, 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?
