Timeline for Trace 0 and Norm 1 elements in finite fields

11 events

when toggle format	what		by	license	comment
May 10, 2017 at 11:45	vote	accept	sampath
May 10, 2017 at 11:45
May 9, 2017 at 14:32	answer	added	Will Sawin		timeline score: 3
May 9, 2017 at 14:12	comment	added	Will Sawin		@sam Certainly not when $q$ is large with respect to $\ell$.
May 9, 2017 at 11:51	comment	added	sampath		@WillSawin, Given $\ell$ and $\zeta$ does there exist $i,j$ with satisfying these equations ?
May 9, 2017 at 11:47	comment	added	sampath		@LSpice, Thank you. What you stated is true.
May 9, 2017 at 2:23	comment	added	LSpice		@KConrad, maybe norm 1 is implicitly referring to the elements $\zeta^{1 - q^i}$?
May 9, 2017 at 1:30	comment	added	Will Sawin		I think it can be shown that for fixed $\ell, i,j$ the number of $\zeta$ satisfying these equations for $q$ large enough is $q^{\ell -2} + O( q^{ \ell - 5/2})$. So the total number of solutions is something like ${\ell -1 \choose 2} q^{\ell -2 } $.
May 9, 2017 at 1:09	comment	added	Will Sawin		Clearly not when $\zeta \in \mathbb F_q$. Also usually not for $\ell$ small, presumably. Or do you mean does there exist some $\zeta$?
May 8, 2017 at 23:45	comment	added	KConrad		The title of the questions mentions trace 0 and norm 1, but the question itself does not mention any norms.
May 8, 2017 at 23:22	history	edited	Gerry Myerson		edited tags
Aug 2, 2016 at 10:22	history	asked	sampath	CC BY-SA 3.0