# Singular

#### D.10.1.10 closed_points

Procedure from library `brnoeth.lib` (see brnoeth_lib).

Usage:
closed_points(I); I an ideal

Return:
list of prime ideals (each a Groebner basis), corresponding to the (distinct affine closed) points of V(I)

Note:
The ideal must have dimension 0, the basering must have 2 variables, the ordering must be lp, and the base field must be finite and prime.
It might be convenient to set the option(redSB) in advance.

Example:
 ```LIB "brnoeth.lib"; ring s=2,(x,y),lp; // this is just the affine plane over F_4 : ideal I=x4+x,y4+y; list L=closed_points(I); // and here you have all the points : L; ==> [1]: ==> _[1]=y ==> _[2]=x ==> [2]: ==> _[1]=y ==> _[2]=x+1 ==> [3]: ==> _[1]=y ==> _[2]=x2+x+1 ==> [4]: ==> _[1]=y+1 ==> _[2]=x ==> [5]: ==> _[1]=y+1 ==> _[2]=x+1 ==> [6]: ==> _[1]=y+1 ==> _[2]=x2+x+1 ==> [7]: ==> _[1]=y2+y+1 ==> _[2]=x+y ==> [8]: ==> _[1]=y2+y+1 ==> _[2]=x+1 ==> [9]: ==> _[1]=y2+y+1 ==> _[2]=x+y+1 ==> [10]: ==> _[1]=y2+y+1 ==> _[2]=x ```