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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?

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I will refrain from opening up the can of worms about the "topological" proof, but there's an algebraic proof due to Larry Washington (and perhaps others before him). A semi-local Dedekind domain is PID (weak approx.), and integral closure of semi-local Dedekind domain in finite sep'ble ext'n of its fraction field is semi-local (only finitely many primes upstairs over a given one downstairs), hence PID. So existence of a number field with class number $> 1$ (e.g., $\mathbf{Q}(\sqrt{-5})$) implies $\mathbf{Z}$ is not semi-local. QED (This is not circular...) –  BCnrd Oct 5 '10 at 2:48
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let $p_1,\ldots,p_N$ be a complete list of primes. Since no $p_i$ divided $p_1\cdots p_N+1$, this can't be a complete list of primes. –  David Hill Oct 5 '10 at 3:08
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Without some further clarification regarding the use of the word "algebraic", I think this question should be closed. –  S. Carnahan Oct 5 '10 at 3:32
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@BCnrd: isn't your separability hypothesis superfluous? (If so, are you feeling okay? Do you require any medical attention??) –  Pete L. Clark Oct 5 '10 at 3:43
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@Cam: no. BCnrd's interpretation made a good answer of a poor question. –  Kevin Buzzard Oct 5 '10 at 9:45
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closed as too localized by Andy Putman, Felipe Voloch, Gerry Myerson, Yemon Choi, Cam McLeman Oct 5 '10 at 4:25

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1 Answer

[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: Prof. D. Lorenzini mentioned it to me and remarked that it 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?

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Washington's proof appears in Ribenboim's book 'The New Book of Prime Number Records'. The citation was: "1980, Washington, L. C. The infinitude of primes via commutative algebra. Unpublished manuscript." –  Ben Linowitz Oct 5 '10 at 4:10
    
Washington's proof also appears in Ben Chastek, Two proofs of the infinitude of primes, math.umn.edu/~garrett/students/reu/benChastek.pdf but Ben doesn't say where he found it. –  Gerry Myerson Oct 5 '10 at 4:29
    
@Gerry: thanks. Chastek's manuscript includes a reference to a book of Ribenboim which is, I believe, an earlier version of the book mentioned in Ben Linowitz's comment above. So he probably got it there. Perhaps "unpublished manuscript" is the best citation I'm going to get. –  Pete L. Clark Oct 5 '10 at 4:47
    
Dear Pete, Have you contacted Larry Washington directly regarding a possible written account of his proof? –  Emerton Oct 16 '10 at 19:18
    
@ME: no, I haven't wanted to bother him. I suppose though that since I am now bothering other people, it may be worth sending such an email... –  Pete L. Clark Oct 16 '10 at 21:14
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