Timeline for Can $P(z)$ have a divisor in a given congruence class?
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| Nov 16, 2022 at 3:08 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Oct 17, 2022 at 3:08 | answer | added | Will Jagy | timeline score: 3 | |
| Oct 17, 2022 at 2:56 | comment | added | Will Jagy | see also Reuschle, archive.org/details/tafelncomplexer00unkngoog/page/6/mode/2up whole $\lambda = 9$ starts page 173. Then $n=63$ starts page 284, goes to 304. Interesting footnotes; note that two cubics are involved, $x^3-21 x -28$ and $x^3 - 21 x +35$ Found online, lmfdb.org/NumberField/… | |
| Oct 17, 2022 at 2:28 | comment | added | Will Jagy | your given polynomial is a bit special, monic cubic with discriminant a square. These are in table B.4, pages521-523, Henri Cohen, A Course in Computational Number Theory. The Galois group is then cyclic.... next are $x^3-3x-1, \; \; \; $ and $x^3-x^2-4x-1$ see zakuski.math.utsa.edu/~jagy/cox_galois_Gaussian_periods.pdf | |
| Oct 16, 2022 at 17:16 | comment | added | Wojowu | I apologize the above ended up being mostly throwing a lot of big words at you! I hope someone else will be able to turn that into a proper answer. Algorithmic algebraic number theory is quite well developed, so it definitely should be possible. | |
| Oct 16, 2022 at 17:15 | comment | added | Wojowu | Characterizing primes dividing values of a polynomial is a deep and complicated subject - it leads quite directly to class field theory as well as its nonabelian variants, tying even to Langlands program. Assuming $P$ is univariate and irreducible, then those primes can be characterized in terms of factorization of primes in the splitting field $K$ of $P$, and the congruence conditions mod $m$ can be seen by considering the extension $K\mathbb Q(\zeta_m)$. Using Chebotarev density theorem you should be able to characterize all the possibilities. | |
| Oct 16, 2022 at 15:02 | history | edited | Bogdan Grechuk | CC BY-SA 4.0 |
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| Oct 16, 2022 at 14:55 | history | asked | Bogdan Grechuk | CC BY-SA 4.0 |