Just curious...

[Kurt Foster <drsardonicus@earthlink.net>]

I ran into a warning and error message with bnfcertify() that piqued  
my curiosity.  I've had Pari simply bail out on bnfcertify() before,  
because the Minkowski bound was too large.  But this time it merely  
issued a warning on the size of the Minkowski bound, and THEN  
(probably fortunately) bailed out because my primelimit wasn't big  
enough.  "VERY long" sounds ominously like "a geological epoch."

I compared the primelimit suggested by the error message to the  
Minkowski bound.  It's a little less than 1/12th as big.  Does this  
correspond to some refinement of the classical result that every ideal  
class contains an integral ideal whose norm is at most the Minkowski  
bound?

Here's the example:

? fqtrn=x^8 + 53816*x^7 + 724255700*x^6 + 5791731704*x^5 +  
10136135782*x^4 + 5793561160*x^3 + 725116628*x^2 + 161416*x + 1;
time = 0 ms.
? polgalois(fqtrn)
time = 4 ms.
%2 = [8, 1, 5, "Q_8(8)"]
? qtrnfield=bnfinit(fqtrn);
time = 17,625 ms.
? qtrnfield.r1
time = 0 ms.
%4 = 8
? mbound=sqrt(qtrnfield.disc)*8!/8^8
time = 0 ms.
%5 = 374404601290.78125000000000000000000000
? bnfcertify(qtrnfield)
  *** bnfcertify: Warning: large Minkowski bound: certification will  
be VERY long.
  *** bnfcertify: not enough precomputed primes, need primelimit ~  
30439469497.
? mbound/30439469497.
time = 0 ms.
%6 = 12.299971302971661970288772145351163772