How to get the extension degree

Question

The real number field cannot express $\sqrt{2}$, so we extend $2$ times to get $Q(\sqrt{2})=a+b\sqrt(2)$. But it cannot express $\sqrt{3}$ still. So we are going to extend it $2$ more times to get $Q(\sqrt{2},\sqrt{3})=a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}$. Then we express it all now after extend $4$ times.

Similarly, how to know the minimum extension degree from the field of real numbers to include

lis1 = {-(-2)^(1/3), 2^(1/3), (-1)^(2/3) 2^(1/3)}

or

lis2 = {-2,
        -(1/4) + (I Sqrt[3])/4 - 1/2 Sqrt[-(1/2) + (7 I Sqrt[3])/2],
        -(1/4) - (I Sqrt[3])/4 - 1/2 Sqrt[-(1/2) - (7 I Sqrt[3])/2],
        -(1/4) - (I Sqrt[3])/4 + 1/2 Sqrt[-(1/2) - (7 I Sqrt[3])/2],
        -(1/4) + (I Sqrt[3])/4 + 1/2 Sqrt[-(1/2) + (7 I Sqrt[3])/2]}

by MMA?

Answer 1

You can use ToNumberField to do this:

extensionDegree[l_List] := Replace[ToNumberField[l, All],
    {
    {___, a_AlgebraicNumber, ___} :> Length @ Last @ a,
    _->Missing["NotAvailable"]
    }
]

Your examples:

extensionDegree[{Sqrt[2], Sqrt[3]}]
extensionDegree[{-(-2)^(1/3), 2^(1/3), (-1)^(2/3) 2^(1/3)}]
extensionDegree[{-2, -(1/4) + (I Sqrt[3])/4 -  1/2 Sqrt[-(1/2) + (7 I Sqrt[3])/2], -(1/4) - (I Sqrt[3])/4 -  1/2 Sqrt[-(1/2) - (7 I Sqrt[3])/2], -(1/4) - (I Sqrt[3])/4 +  1/2 Sqrt[-(1/2) - (7 I Sqrt[3])/2], -(1/4) + (I Sqrt[3])/4 +  1/2 Sqrt[-(1/2) + (7 I Sqrt[3])/2]}]

4

6

8