Timeline for Factoring a certain quartic mod primes
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 15, 2012 at 12:57 | vote | accept | Richard | ||
| Mar 15, 2012 at 10:30 | comment | added | Geoff Robinson | It might be easier to set $y = x+3$ and work with the polynomial $y^4 - 40y^2 + 120y -80.$ | |
| Mar 15, 2012 at 6:31 | answer | added | Zack Wolske | timeline score: 1 | |
| Mar 15, 2012 at 6:00 | answer | added | Noam D. Elkies | timeline score: 5 | |
| Mar 15, 2012 at 5:13 | comment | added | Franz Lemmermeyer | This was also posted on math stackexchange. | |
| Mar 15, 2012 at 0:43 | answer | added | Aaron Meyerowitz | timeline score: 2 | |
| Mar 14, 2012 at 23:10 | comment | added | Zack Wolske | It seems the congruences you want are $p \equiv \pm 11 \mod 30$, and $p \equiv \pm 1 \mod 30$. The other cases $(7,13,17,23)$ are taken care of. | |
| Mar 14, 2012 at 23:02 | comment | added | Zack Wolske | $2$ is a root mod $29$: $16 + 96 + 56 - 24 + 1 = 88 + 57 = 87 + 58 = 3(29) + 2(29)$, so $h$ has a linear factor. | |
| Mar 14, 2012 at 21:43 | answer | added | Will Jagy | timeline score: 4 | |
| Mar 14, 2012 at 19:18 | comment | added | Richard | The extension determined by h over Q is Galois, and SAGE computes it as C4, the cyclic group of order 4, not as the Klein 4 group. By the way, this isn't from a homework question. I'm using it to determine when a parameterized family of elliptic curves has j-invariant 0. | |
| Mar 14, 2012 at 18:52 | answer | added | user19475 | timeline score: 2 | |
| Mar 14, 2012 at 18:47 | comment | added | user19475 | MAGMA calculated the Klein four group $\mathbf{Z}/2 \times \mathbf{Z}^2$. Now use class field theory and Chebotarev. | |
| Mar 14, 2012 at 18:26 | comment | added | user19475 | Can you tell us the Galois group of $h$ over $\mathbf{Q}$? It must be abelian if the splitting is determined by congruence conditions. | |
| Mar 14, 2012 at 18:23 | comment | added | Igor Rivin | This has a homework scent to it... | |
| Mar 14, 2012 at 18:20 | history | asked | Richard | CC BY-SA 3.0 |