Timeline for Does the expression $x^4 +y^4$ take on all values in $\mathbb{Z}/p\mathbb{Z}$?
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Apr 6, 2020 at 8:00 | vote | accept | Junsukim | ||
| Apr 6, 2020 at 8:00 | vote | accept | Junsukim | ||
| Apr 6, 2020 at 8:00 | |||||
| Apr 2, 2020 at 11:10 | vote | accept | Junsukim | ||
| Apr 6, 2020 at 8:00 | |||||
| Apr 1, 2020 at 19:08 | comment | added | Robert Israel | See this question. | |
| Apr 1, 2020 at 18:20 | answer | added | Joe Silverman | timeline score: 12 | |
| Apr 1, 2020 at 17:23 | answer | added | R.P. | timeline score: 6 | |
| Apr 1, 2020 at 16:35 | history | edited | R.P. | CC BY-SA 4.0 |
added 101 characters in body; edited title
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| Apr 1, 2020 at 16:25 | comment | added | Junsukim | @Daniel could you explain why the answer is yes in that case please? | |
| Apr 1, 2020 at 16:23 | comment | added | Junsukim | Ah, I'm sorry. What I intended was $\mathbb{Z}/ p\mathbb{Z}$ , not p-adic integers | |
| Apr 1, 2020 at 15:20 | comment | added | Daniel Loughran | So are you clarifying that $\mathbb{Z}_p$ does indeed denote the $p$-adic integers? | |
| Apr 1, 2020 at 15:10 | comment | added | Franka Waaldijk | @Daniel ah i forgot that case, very silly of me, thank you for correcting me. | |
| Apr 1, 2020 at 14:44 | answer | added | emtom | timeline score: 5 | |
| Apr 1, 2020 at 14:34 | comment | added | Daniel Loughran | Actually it is quite important whether $\mathbb{Z}_p$ means the $p$-adic integers or the finite field. In the latter case the answer is yes, but not in the former case. The problem is that $x^4 + y^4 = p$ has a solution in the $p$-adics if and only if $p$ is split in the $8$th cyclotomic field (i.e. $p \equiv 1 \bmod 8$). The easiest way I know how to prove it over a finite field is via the Hasse-Weil theorem, but probably there is an elementary approach. | |
| Apr 1, 2020 at 13:43 | comment | added | Franka Waaldijk | Ok, I get you now :-) | |
| Apr 1, 2020 at 13:43 | comment | added | YCor | @FrankaWaaldijk sure but it would matter in the way to formulate an answer. | |
| Apr 1, 2020 at 13:39 | comment | added | Franka Waaldijk | @YCor I'm not sure that matters, in the light of Hensel's lemma...by which I suspect that if the statement holds for $\mathbb{Z}/p\mathbb{Z}$, then it holds for $\mathbb{Z}_p$. | |
| Apr 1, 2020 at 13:32 | comment | added | YCor | I'm not sure whether you're using $\mathbb{Z}_p$ to denote the ring of $p$-adics or for $\mathbf{Z}/p\mathbf{Z}$. | |
| Apr 1, 2020 at 13:31 | history | edited | YCor | CC BY-SA 4.0 |
edited title
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| Apr 1, 2020 at 13:12 | history | asked | Junsukim | CC BY-SA 4.0 |