Timeline for On Cubic Non-Residues Modulo a Prime
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 10, 2013 at 20:25 | vote | accept | Turbo | ||
| Jul 10, 2013 at 19:34 | comment | added | Turbo | Ok. Thankyou very much. | |
| Jul 10, 2013 at 19:33 | comment | added | David E Speyer | Sorry, my opinion is that I don't know. If you held $M$ and $N$ fixed and looked at primes between $T$ and $2T$, you'd be right, but I don't know enough about the error bounds in Cebatarov to know what happens when you link the endpoints of your interval to the cubic extension. But there are plenty of experts here! | |
| Jul 10, 2013 at 19:31 | comment | added | Turbo | I am guessing the answer is same for the first question as well as $M,N \rightarrow \infty$ since the number of primes to choose from in the middle increases linearly in $M$ and $N$. What is your opinion? | |
| Jul 10, 2013 at 19:30 | comment | added | David E Speyer | Yes, that's it. | |
| Jul 10, 2013 at 19:29 | comment | added | Turbo | Oh I see. Total #of $\epsilon_{p}$ is $2$ (which counts $\pm $), total number of $a_{p},b_{p}$ each is $3$ (which counts $0,1,2$). So total possibilities is $18$. Of which we want $\epsilon_{p}=1$, $a_{p}=0$ and $b_{p}\in \{1,2\}$. This counts to $2/18$. Correct? | |
| Jul 10, 2013 at 19:28 | comment | added | David E Speyer | To your first question, I don't know. To the second question: Cebatarov density states that, for $C$ a conjugacy closed subset of a Galois group $G$, the probability that $Frob(p)$ will be in $C$ is $|C|/|G|$. In this case, the Galois group has $18$ elements and $2$ of them work. For the Cebatarov density theorem, see an "advanced" textbook in algebraic number theory, such as Janusz (Algebraic Number Fields) or Neukirch (Algebraic Number Theory). | |
| Jul 10, 2013 at 19:24 | comment | added | Turbo | I am unfamiliar with the density theorem. Could you provide the details for $2/18$ step? | |
| Jul 10, 2013 at 19:07 | comment | added | Turbo | Thankyou. Does the same probability hold if the prime has to be between $M$ and $N$ if $N < 2M$ or $M < 2N$? | |
| Jul 10, 2013 at 18:56 | history | answered | David E Speyer | CC BY-SA 3.0 |