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multiplication matrix in number field

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? ?

multiplication matrix in number field

Hello 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 . Then, I don't know how to do these calculations automatically within SAGE:

=> for example: how make the vector 1=[1,0,0,0,0,0]a²=[0,0,1,0,0,0] become [0,1,0,1,0,0](a+b).a²=2+a²b=[2,0,0,,0,1] after multiplication of by a+b given the relations a³=2 and b²=3? ?

multiplication matrix in number field

Hello 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 row 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() . Then, I don't know how to do these calculations automatically within SAGE:

=> for example: how make the vector a²=[0,0,1,0,0,0] become (a+b).a²=2+a²b=[2,0,0,,0,1] after multiplication of by a+b given the relations a³=2 and b²=3? ?

multiplication matrix in number field

Hello 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 row column is the image of each element times a+b: (transpose the vectors bellow)

• 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() . Then, I don't know how to do these calculations automatically within SAGE:

=> for example: how make the vector a²=[0,0,1,0,0,0] become (a+b).a²=2+a²b=[2,0,0,,0,1] after multiplication of by a+b given the relations a³=2 and b²=3? ?