Kurt Foster on Wed, 30 Apr 2008 05:53:05 +0200

 Just curious...

• To: Pari Users <pari-users@list.cr.yp.to>
• Subject: Just curious...
• Date: Tue, 29 Apr 2008 21:37:18 -0600
• Delivery-date: Wed, 30 Apr 2008 05:53:05 +0200
• Mailing-list: contact pari-users-help@list.cr.yp.to; run by ezmlm

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

```