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

[Bill Hale <hale@tulane.edu>]

At 10:23 PM -0400 6/14/03, Mak Trifkovic wrote:
>Hi,
>
>I need help with the following:
>
>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]?
>
>thanks,
>Mak

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 output that I get for last three statements is:

? %5 = [1, y]
? %6 = [Mod(1, y^2 + y + 1), Mod(1, y^2 + y + 1)*x, Mod(1, y^2 + y +
1)*x^2, Mod(1, y^2 + y + 1)*x^3, Mod(1, y^2 + y + 1)*x^4, Mod(1, y^2 + y +
1)*x^5]
? %7 = [1, x, x^2, x^3, x^4, x^5]


-- Bill Hale