Timeline for Is there an "elementary" proof of the infinitude of completely split primes?
Current License: CC BY-SA 2.5
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| Jan 15, 2022 at 20:25 | comment | added | KConrad | @FelipeVoloch your link above no longer works. Since you didn't mention bibliographic information, here it is for anyone else: "Chebyshev's method for number fields" Journal de Théorie des Nombres de Bordeaux 12 (2000), 81-85. A working link (for now) is numdam.org/article/JTNB_2000__12_1_81_0.pdf. | |
| Apr 24, 2015 at 9:59 | comment | added | Ofir Gorodetsky | You can actually consider $\phi_n(m p_1 \cdots p_r)$ where $m$ is the smallest prime divisor of $n$. The reason is that prime divisors of $\phi_n(x)$ are either $1 \mod n$, or $m$. | |
| Feb 14, 2010 at 16:13 | comment | added | Victor Miller | @Felipe: thanks for the reference to your paper. Very nice! That's what I was after. And I also liked your statement "Let's stop pretending that we don't know anything about the distribution of splitting primes". | |
| Feb 14, 2010 at 15:51 | comment | added | Felipe Voloch | I wrote a note some time ago extending Chebyshev's method to number fields which gives an elementary and simple proof that the number of split primes is at least $x^{1/d}/\log x$. jtnb.cedram.org/jtnb-bin/fitem?id=JTNB_2000__12_1_81_0 | |
| Feb 14, 2010 at 2:27 | history | edited | Victor Miller | CC BY-SA 2.5 |
added 19 characters in body
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| Feb 14, 2010 at 2:10 | comment | added | Victor Miller | So, it does bring up the question -- how dense a set of splitting primes can we construct with an elementary argument? The set that "Euclid-like" arguments constructs is awfully sparse. | |
| Feb 14, 2010 at 2:07 | comment | added | Victor Miller | As I was typing my answer it occurred to me that the same idea gave something like Bjorn's answer, but I wanted to get it in quickly :-). | |
| Feb 14, 2010 at 2:04 | comment | added | Bjorn Poonen | Taking $\alpha$ to be a primitive n-th root of 1 in my answer gives your answer for that case! | |
| Feb 14, 2010 at 1:50 | history | answered | Victor Miller | CC BY-SA 2.5 |