Karim BELABAS on Sun, 15 Jun 2003 14:25:00 +0200 (MEST)

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 Re: Finding an integral basis of a relative extension of nf's

```On Sun, 15 Jun 2003, Bill Hale wrote:
> At 10:23 PM -0400 6/14/03, Mak Trifkovic wrote:
>>I have a number field K=bnfinit(P(y)), and an extension L given by a
>>polynomial Q(x) in K[x].  Assuming K has class number one,how do I get
>>gp to find an integral basis for O_L over O_K, expressed as a set of
>>polynomials in x with coefficients in Q[y]?
> I am new to Pari myself. I am not sure if the following is
> totally correct. Look at the following and refer to the
> manual for details. You may wish to try your own polynomials
> to test.
>
> ==================================
> py=y^2+y+1
> qx=x^6+x^5+x^4+x^3+x^2+x+1
> K=bnfinit(py)
> L=rnfinit(K,qx)
> K.zk
> L[7][1]
> lift(L[7][1])

The above is incorrect in general: it assumes the ideals in L[7][2] are all
equal to O_K, i.e represented by identity matrices (which is indeed true in
thsi specific case.

[ Background: L[7] is only a pseudo-basis (see rnfpseudobasis()) for O_L, not
a true O_K-basis, since the latter does not exist in general. And can be much
more expensive to compute even if it does exist ].

What you're looking for is

rnfbasis(bnfinit(py), qx)

Cheers,

Karim.
--
Karim Belabas                     Tel: (+33) (0)1 69 15 57 48
Dép. de Mathématiques, Bât. 425   Fax: (+33) (0)1 69 15 60 19
Université Paris-Sud              http://www.math.u-psud.fr/~belabas/
F-91405 Orsay (France)            http://www.parigp-home.de/  [PARI/GP]
```