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`

? ?