On Mon, Apr 14, 2008 at 12:23 PM, Craig Citro <[EMAIL PROTECTED]> wrote: > > Actually, Conway polynomials don't cut it in general, in the sense > that one can't actually generate them for sufficiently large finite > fields. Even Magma doesn't know them in the sizes that Willem is > talking about: > > > ConwayPolynomial(3,100); > ^ > Runtime error in 'ConwayPolynomial': A conway polynomial for GF(3^100) is not > known > > David Roe and I (and probably some other people) talked about this way > back at SD3, and ultimately decided that we had no idea what Magma was > doing. ;) David (and/or others) may have discovered more by now. Magma > definitely has a way on deciding on a default choice, in the sense > that if you create the finite fields in the same order, the same > polynomial is generated between different Magma runs. It's interesting > that it changes when you change the order you created the fields with > -- I don't recall us discovering this way back when. I also find this > somewhat interesting -- though it's probably just the fact that it's > checking something for consistency: > > [EMAIL PROTECTED] ~] $ magma > Magma V2.13-5 Mon Apr 14 2008 09:21:46 on sage [Seed = 391961513] > Type ? for help. Type <Ctrl>-D to quit. > > time F := FiniteField(3^100); > Time: 0.050 > > time K := FiniteField(3^200); > Time: 3.470 > > > > Total time: 4.190 seconds, Total memory usage: 8.39MB > [EMAIL PROTECTED] ~] $ magma > Magma V2.13-5 Mon Apr 14 2008 09:22:08 on sage [Seed = 508940938] > Type ? for help. Type <Ctrl>-D to quit. > > time K := FiniteField(3^200); > Time: 0.420 > > time F := FiniteField(3^100); > Time: 0.050 > > > > That paper is particularly frustrating -- if you're trying to figure > out exactly this, you end up getting referred around the paper a few > times, only to discover a paragraph where they say "and in the case > that the Conway polynomial can't be found, we make another default > choice." Wow, thanks. :) > > -cc > > > > > >
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