> with(linalg):

Here we study computations in the field Q(a=sqrt(2),b=sqrt(3)). Let M be the submodule spanned by

1, a and N that spanned by 1, b. Then MN is spanned by 1, a, b, ab. First we write the matrices

corresponding to the action of a and b respectively on this generating set.

a:=matrix(4,4,[[0,1,0,0],[2,0,0,0],[0,0,0,1],[0,0,2,0]]);

> b:=matrix(4,4,[[0,0,1,0],[0,0,0,1],[3,0,0,0],[0,3,0,0]]);

Now we can calculate the characteristic polynomial of the sum and product of a and b.

> evalm(a+b);

>

>

> charpoly(%,x);

We can verify that this polynomial has a+b as a root:

> subs(x=sqrt(2)+sqrt(3),%);

> simplify(%);

Now we can work out the case of ab:

> multiply(a,b);

> charpoly(%,t);

>

> subs(t=sqrt(6),%);