Gerhard Niklasch on Sat, 5 Dec 1998 01:59:18 +0100 (MET) |
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bug in polredabs() (fwd) |
In response to > Message-Id: <19981204193944.K14565@io.txc.com> > Date: Fri, 4 Dec 1998 19:39:44 -0500 > From: Igor Schein <igor@txc.com> [...] > So x^16+48 and x^16+3 generate the same number field, No, they don't define the same field. They define two distinct fields which happen to have the same discriminant 2^48*3^15. (If they did, then the field would also contain a 16th root of (-48)/(-3) = 16, or in other words, a 4th root of 2. But it doesn't even contain a square root of 2 -- try to nffactor x^2-2 over the result of nfinit(polredabs(y^16+3)). The fields are quite hard to tell apart otherwise -- a quick glance at the first 160 coefficients of their Dedekind zeta functions suggests that they may be arith- metically equivalent... any Galois representation experts at hand? I'm afraid I've had too long a week at work to check this.) That x^16+48 is difficult to work with is not surprising -- the discriminant of the polynomial is divisible by a large power of 3 and by a huge power of 2, and the integer-base algorithm has to figure out how much of each is due to the field discriminant and how much to the index... x^16-12*x^8+48 is marginally better (2^108*3^15 instead of 2^124*3^15), but still a lot harder than x^16+3 with polynomial discriminant 2^64*3^15. Cheers, Gerhard