zekhnini on Fri, 06 Jan 2012 23:19:00 +0100

 Re: ideal principal in relative extension

```zekhnini <zekha2 <at> gmail.com> writes:

>
>
> Dear users,
>  I want to know if the thow ideals M and N are principal in the relative
> extension K=k(sqrt(2+8*i)) of the field k=Q(sqrt(2*17*5),i ). what can I do?
>
>
>   T=polcompositum(y^2+1,y^2-2*17*5)[1];
>   k=nfinit(T);
>   P=x^2-Mod(y^4-4*y^2+4*17,k.pol);
>
>   K=rnfinit(k,P);
>   Keqn=rnfequation(k,P,1)[1];
>   Kabs=nfinit(Keqn);
>
>   H1=idealprimedec(k,17)[1];
>   H2=idealprimedec(k,17)[2];
>
>   M=rnfidealup(K,H1);
>   N=rnfidealup(K,H2);
>
>   MK= mathnf( Mat( nfalgtobasis(Kabs, M) ) );
>   NK= mathnf( Mat( nfalgtobasis(Kabs, N) ) );
>
>  It responds  nfalgtobasis: incorrect type in algtobasis.
>
> Thanks for help.
> zekhnini
>
>
Vous trouvez ci-dessus la rÃponse de Mr Kurt Foster :

The following computations may be useful.  Note in particular that defining
polynomials are produced using rnfequation() and then reduced using polredabs
() .  The command z;y;t; was the very first in the session, and set the
priority of the variable t to be lower than that of y, and the priority of y to
be lower than that of z.  The command rnfequation() does not require number
field computations, but is  fussy about variable priority.  The variable x
always has the highest priority.

I used the command galoisinit(T3) simply to check that the polynomial T3 is a
defining polynomial for a normal extension of Q  (the degree 16 is too high to
use polgalois()).  If it hadn't been, galoisinit() would have complained.

The polynomial P defines a non-normal quartic extension k of Q.  The
polynomial T defines the field you asked about, which is the join of k with a
quadratic field.  I called it F.

The polynomial T2 is a defining polynomial for the normal closure L of k over
Q.  Note that the prime ideals lying over 17 in L are all principal, even
though L has class group C2 x C2.

The polynomial T3 is a defining polynomial for the normal closure L3 of F over
Q.  The prime ideals lying over 17 in L3 are all principal, though L3 has class
group C12 x C2.  The two non-principal ideals lying over 17 in F remain prime
and become principal in L3.  The principal ideal in F3 lying over 17 splits
into two principal ideals in L3.

? z;y;t;

? p1=t^2+1;p2=y^2-2-8*t;p3=z^2-2*5*17;

? P=polredabs(rnfequation(p1,p2))
%3 = y^4 - 8*y^2 + 17

? T=polredabs(rnfequation(P,p3))
%4 = z^8 - 4*z^7 - 14*z^6 + 56*z^5 + 163*z^4 - 424*z^3 + 1042*z^2 - 820*z + 746

? F=bnfinit(T);

? F.clgp
%6 = [24, [12, 2], [[26, 21, 1, 5, 16, 19, 16, 19; 0, 1, 0, 0, 0, 0, 0, 0; 0,
0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 2, 1, 1, 1; 0, 0, 0,
0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 1], [10, 4, 4, 5,
1, 4, 5, 3; 0, 2, 0, 1, 1, 1, 1, 1; 0, 0, 2, 1, 0, 0, 0, 1; 0, 0, 0, 1, 0, 0,
0, 0; 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0;
0, 0, 0, 0, 0, 0, 0, 1]]]

? vf=idealprimedec(F,17)
%7 = [[17, [-4, 0, 0, 1, 0, 1, 0, 0]~, 2, 1, [-2, 8, -5, 3, -1, 4, -6, 7]~],
[17, [3, 0, 0, 1, 0, 1, 0, 0]~, 2, 1, [-2, 8, 5, -3, -1, 4, 6, -7]~], [17, [7,
-6, 0, 1, 0, 0, -2, 0]~, 2, 2, [0, 0, -1, -4, 0, 0, 8, -2]~]]
^---
? for(i=1,#vf,print(bnfisprincipal(F,vf[i])))

[[6, 1]~, [1198557/96536180, 883157/193072360, -23461/96536180,
-2958581/193072360, 1602329/193072360, -446233/193072360, 406537/193072360,
1140261/193072360]~]
[[0, 1]~, [1/10, -17/10, 7/10, 1/10, -2/5, 3/10, -1/5, -1/10]~]
[[0, 0]~, [0, 0, 1, 0, 0, 0, 0, 0]~]

? k=bnfinit(P);

? k.clgp
%9 = [2, [2], [[5, 3, 4, 3; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1]]]

? vk=idealprimedec(k,17)
%10 = [[17, [-5, 0, 1, 0]~, 1, 1, [3, 5, 4, 1]~], [17, [0, 0, 1, 0]~, 2, 1,
[0, 0, -4, 1]~], [17, [5, 0, 1, 0]~, 1, 1, [-3, -5, 4, 1]~]]

? for(i=1,#vk,print(bnfisprincipal(k,vk[i])))
[[0]~, [-1, 1, 0, -1]~]
[[0]~, [0, 0, 0, -1]~]
[[0]~, [1, -1, 0, -1]~]

? T2=polredabs(rnfequation(P,x^2-17))
%11 = x^8 + 9*x^4 + 16

? L=bnfinit(T2);

? L.clgp
%13 = [4, [2, 2], [[13, 8, 8, 11, 3, 12, 7, 9; 0, 1, 0, 0, 0, 0, 0, 0; 0, 0,
1, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0,
0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 1], [13, 8, 10, 1, 5,
2, 5, 12; 0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0,
0; 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0; 0,
0, 0, 0, 0, 0, 0, 1]]]

? vl=idealprimedec(L,17)
%14 = [[17, [-7, 0, 1, 0, 0, 0, 0, 0]~, 2, 1, [7, 6, 1, -4, -1, 4, 3, 5]~],
[17, [-6, 0, 1, 0, 0, 0, 0, 0]~, 2, 1, [-6, -7, -1, 4, -1, 4, -5, -3]~], [17,
[6, 0, 1, 0, 0, 0, 0, 0]~, 2, 1, [6, 7, -1, 4, -1, 4, 5, 3]~], [17, [7, 0, 1,
0, 0, 0, 0, 0]~, 2, 1, [-7, -6, 1, -4, -1, 4, -3, -5]~]]

? for(i=1,#vl,print(bnfisprincipal(L,vl[i])))
[[1, 1]~, [-4/13, -6/13, -3/13, 5/13, 1/13, -2/13, 2/13, 3/13]~]
[[1, 1]~, [7/13, 4/13, 5/13, -2/13, -3/13, -1/13, -2/13, -3/13]~]
[[1, 1]~, [6/13, -4/13, -3/13, 0, 0, -2/13, 3/13, -2/13]~]
[[1, 1]~, [2/13, 3/13, 1/13, -2/13, -3/13, 5/13, 2/13, 3/13]~]

? T3=polredabs(rnfequation(T,x^2+x-4))
%15 = x^16 - 4*x^14 + 78*x^12 + 384*x^10 + 433*x^8 - 5052*x^6 + 10212*x^4 -
9472*x^2 + 4096

? gf=galoisinit(T3);

? L3=bnfinit(T3);

? L3.clgp
%18 = [24, [12, 2], [[26, 5, 5, 14, 24, 21, 22, 15, 10, 19, 23, 5, 9, 23, 1,
15; 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 2, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1; 0, 0, 0,
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0,
1, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0; 0,
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,
0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 1], [13, 5, 0, 3, 12, 10, 10, 2, 5, 1, 4, 8, 8, 1, 11, 5; 0,
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 13, 12, 8, 8, 0, 12, 11, 10,
2, 10, 2, 11, 8, 1; 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0,
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0; 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0; 0, 0,
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,
0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 1]]]

? vl3=idealprimedec(L3,17)
%19 = [[17, [-5, 4, -8, 8, -8, 0, 0, 0, 0, 1, -2, 0, -2, -1, 0, 0]~, 2, 2, [0,
1, -6, 8, -2, -2, -5, -3, 4, 2, -1, 0, -4, -1, -3, 5]~], [17, [-5, 4, 8, -8, 8,
0, 0, 0, 0, 1, -2, 0, -2, -1, 0, 0]~, 2, 2, [0, 1, 6, -8, 2, 2, 5, 3, 4, 2, -1,
0, -4, -1, 3, -5]~], [17, [8, 4, -5, 5, -5, 0, 0, 0, 0, 1, -2, 0, -2, -1, 0,
0]~, 2, 2, [-8, -2, 5, -4, -1, 7, -1, 6, -1, -5, 4, -3, -4, -1, 5, -3]~], [17,
[8, 4, 5, -5, 5, 0, 0, 0, 0, 1, -2, 0, -2, -1, 0, 0]~, 2, 2, [-8, -2, -5, 4, 1,
-7, 1, -6, -1, -5, 4, -3, -4, -1, -5, 3]~]]

? for(i=1,#vl3,print(bnfisprincipal(L3,vl3[i])))
[[0, 0]~, [-4, -5, 4, -5, -1, -3, 3, 0, 1, 0, 2, 0, -2, 3, 1, -1]~]
[[0, 0]~, [-1, 0, 2, -2, -4, 0, 2, -1, 1, 0, 0, 0, -2, 1, 1, -1]~]
[[0, 0]~, [-4, 0, 5, -1, 3, 0, -4, -4, 0, -1, -1, 3, 0, -1, 1, -1]~]
[[0, 0]~, [7, -8, -4, 7, -2, -3, 14, 5, -2, 0, 3, -2, -5, -6, -5, -1]~]

Thanks for Kurt Foster

```