Given two integral numbers, say a and b such that a^3=2 and b^2=3, I want to check "by hand" that a+b is again integral over Z. To do that, I want to build the matrix of the linear transform "m: multiplication by a+b on the number field Q(a,b)
since a+b is obviously an eigenvalue, i.e. root of it's caracteristic polynomial which then SAGE can compute for me). Then I can check that (a+b).minpoly() gives the same result as P(X)=(M-XI).determinant() and then after coercision within Q.<x>=QQ[], P=Q(P), check P.is_irreducible(), meaning a+b integral.
Thus, given a basis (1,a,a²,b,ab,a²b) for Q(a,b), one can determine the matrix "by hand": each column is the image of each element times a+b: (transpose the following rows in columns)
- 1st is:
(a+b).1=[0,1,0,1,0,0], - 2d is
(a+b).a=[0,0,1,0,1,0], - 3d is
(a+b).a²=[2,0,0,0,0,1], - 4th is
(a+b).b=[3,0,0,0,1,0], - 5th is
(a+b).ab=[0,3,0,0,0,1], - 6th is
(a+b).a²b=[0,0,3,2,0,0]
One can define the extension k.<a,b>=NumberField([x³-2,x²-3]), isomorphic to z⁶-9*z⁴-4*z³+27*z²-36*z-23 (via k.absolute_field() but then I don't know how to do these calculations automatically within SAGE:
how make the vector 1=[1,0,0,0,0,0] become [0,1,0,1,0,0] after multiplication by a+b given the relations a³=2 and b²=3? ?
