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First we compute the "norm form", or the norm of a general element in the field Q(sqrt(2),sqrt(3)),relative

to the basis 1, sqrt(2), sqrt(3), sqrt(6). First the matrices representing sqrt(2), sqrt(3), 1, and sqrt(6)

> 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]]);

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

> z:=multiply(a,b);

Need to load the linear algebra package.

> with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

> z:=multiply(a,b);

Now the matrix of a generic element in terms of the basis

> evalm(x1*e+x2*a+x3*b+x4*z);

Norm of an element is the determinant of the representing matrix (by definition):

> det(%);

Note that the trace of the generic element is 4x1.

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