# Singular

#### D.15.28.1 nfmodSyz

Procedure from library `nfmodsyz.lib` (see nfmodsyz_lib).

Usage:
nfmodSyz(I); I ideal or module

Return:
syzygy module of I over an algebraic number field

Example:
 ```LIB "nfmodsyz.lib"; ring r1 =(0,a),(x,y),(c,dp); minpoly = (a^3+2a+7); module M1 = [(a/2+1)*y, 3*x-a*y], [y-x,y2], [x2-xy, ax-y]; nfmodSyz(M1); ==> _[1]=[x2y2-xy3+(a)*x2+(-a-1)*xy+y2,-3*x3+(a+3)*x2y+(-a)*xy2+(1/2a2+a)*xy+\ (-1/2a-1)*y2,(-1/2a-1)*y3-3*x2+(a+3)*xy+(-a)*y2] ring r2 = (0,a),(x,y,z),(dp,c); minpoly = (a3+a+1); module M2 = [x2z+x+(-a)*y,z2+(a+2)*x], [y2+(a)*z+(a),(a+3)*z3+(-a)*x2], [-xz+(a2+3)*yz,xy+(a2)*z]; nfmodSyz(M2); ==> _[1]=x2z4*gen(3)+(3/29a2-9/29a+1/29)*x4z*gen(3)+(-1/29a2+3/29a-10/29)*x3y\ z*gen(2)+xz4*gen(1)+(-a2-3)*yz4*gen(1)+(1/29a2-3/29a+10/29)*xy3*gen(1)+(3\ /29a2-9/29a+1/29)*x3z*gen(1)+(-7/29a2+21/29a-12/29)*x2yz*gen(1)+(-9/29a2-\ 2/29a-3/29)*x2z2*gen(2)+(-1/29a2+3/29a-10/29)*y2z2*gen(3)+(-1/29a2+3/29a-\ 10/29)*xz3*gen(2)+xz3*gen(3)+(12/29a2-7/29a+33/29)*yz3*gen(2)+(-a)*yz3*ge\ n(3)+(3/29a2-9/29a+1/29)*x3*gen(3)+(-1/29a2+3/29a-10/29)*x2y*gen(2)+(9/29\ a2+2/29a+3/29)*x2y*gen(3)+(-3/29a2+9/29a-1/29)*xy2*gen(2)+(1/29a2-3/29a-1\ 9/29)*xy2*gen(3)+(1/29a2-3/29a-19/29)*x2z*gen(2)+(-3/29a2+9/29a-1/29)*xyz\ *gen(1)+(17/29a2+7/29a+54/29)*xyz*gen(2)+(9/29a2+2/29a+3/29)*y2z*gen(1)+(\ 3/29a2-9/29a+1/29)*z3*gen(3)+(-3/29a2+9/29a-1/29)*xy*gen(1)+(-9/29a2-2/29\ a-3/29)*xz*gen(2)+(-3/29a2-20/29a-1/29)*xz*gen(3)+(2/29a2-6/29a-9/29)*yz*\ gen(2)+(2/29a2-6/29a-9/29)*z2*gen(1)+(3/29a2-9/29a+1/29)*z2*gen(3)+(-3/29\ a2-20/29a-1/29)*x*gen(3)+(2/29a2-6/29a-9/29)*z*gen(1) ring r3=0,(x,y),dp; // ring without parameter module M3 = [x2 + y, xy], [-7y, 2x], [x2-y, 0]; nfmodSyz(M3); ==> _[1]=x2y*gen(2)+2x2*gen(3)-2x2*gen(1)+7y2*gen(3)-y2*gen(2)+2y*gen(3)+2y*g\ en(1) ring r4=0,(x,y),(c,dp); // ring without parameter module M4 = [xy, x-y], [x2 + y, 5y], [- 7y, 2x], [x2-y, 0]; nfmodSyz(M4); ==> _[1]=[0,x3-xy,-5/2x2y+5/2y2,-x3-xy-35/2y2] ==> _[2]=[x+35/4y,-1/2x2-7/4x+7/4y,5/4xy-1/2x+1/2y,1/2x2+7/4x-7/4y] ==> _[3]=[y2-16/1225y,-2/35x2y+156/6125x2-53/245xy+1/5y2-16/6125y,1/7xy2-78/1\ 225xy+2/49y2+8/1225y,2/35x2y-156/6125x2+53/245xy-3/35y2-296/6125y] ```