Is there an algebraic proof of the infinitude of primes? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-25T17:54:55Z http://mathoverflow.net/feeds/question/41096 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41096/is-there-an-algebraic-proof-of-the-infinitude-of-primes Is there an algebraic proof of the infinitude of primes? Mike Jones 2010-10-05T02:35:08Z 2010-10-05T04:06:33Z <p>It is well-known that there exists a (justly celebrated) topological proof of the infinitude of primes (Hillel Fürstenburg, 1955). Does there also exist an algebraic proof?</p> http://mathoverflow.net/questions/41096/is-there-an-algebraic-proof-of-the-infinitude-of-primes/41099#41099 Answer by Pete L. Clark for Is there an algebraic proof of the infinitude of primes? Pete L. Clark 2010-10-05T03:05:28Z 2010-10-05T03:48:36Z <p>[I don't really know what constitutes an "algebraic proof" of infinitude of prime numbers.]</p> <p>The proof that BCnrd alludes to above is described in somewhat more length on p. 5 of</p> <p><a href="http://math.uga.edu/~pete/4400primes.pdf" rel="nofollow">http://math.uga.edu/~pete/4400primes.pdf</a></p> <p>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?</p> <p>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: Prof. D. Lorenzini mentioned it to me and remarked that it had given him some pause. </p> <blockquote> <p>Show that an infinite commutative ring $R$ with finite unit group $R^{\times}$ has infinitely many maximal ideals. </p> </blockquote> <p>As Mr. Bill Dubuque pointed out on another forum, this problem goes back at least as far as Kaplansky's <em>Commutative Rings</em> book. He also remarked that it is no harder to prove a slight generalization: if <code>$\# R &gt; \max(\aleph_0, \# R^{\times})$</code>, then $R$ has infinitely many maximal ideals.</p> <p>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?</p>