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Given a polynomial $a(t)\in F_2[t]$, for which irreducible $f(t)\in F_2[t]$ does the trace of $a(t)\bmod f(t)$ equal 0/1? By Artin reciprocity applied to the extension defined by $x^2-x-a(t)$ it should depend only on the residue of $f(t)$ modulo some conductor, but what does this reciprocity law look like explicitly for this case?

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Is math.uconn.edu/~kconrad/blurbs/ugradnumthy/QRchar2.pdf of any help? (Sorry, I have absolutely no time to check myself.) –  darij grinberg Oct 27 '10 at 16:32
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Gee Darij, I hope it is. :) –  KConrad Oct 27 '10 at 19:09

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