Factoring a polynomial over a number field

Question

Consider a polynomial x^3-x-1 and let $\alpha$, $\beta$, $\gamma$ be three zeros of the polynomial where $\alpha \in \mathbb{R}$. Since $\beta \not \in \mathbb{Q}[\alpha]$ and the degree of the splitting field of the polynomial over $\mathbb{Q}$ is $6$, the minimal polynomial of $\beta$ over $\mathbb{Q}[\alpha]$ should be of degree $2$. To find the polynomial (by the way, I know it should be $x^2+\alpha x+ \frac{1}{\alpha}$.) I have coded the following

sol = x /. Solve[x^3 == x + 1, x];
al = sol[[1]];
be = sol[[2]];
ga = sol[[3]];
MinimalPolynomial[be, x, Extension -> al]

However, Mathematica returns an unexpected message;

What is wrong with the code? What would be a right way to get correct answer?

Answer 1

MinimalPolynomial[s, x, Extension->a] finds the characteristic polynomial of the element $s \in \mathbb{Q}[a]$ over the field $\mathbb{Q}[a]$.

https://planetmath.org/characteristicpolynomialofalgebraicnumber

As has been already suggested, you can get the minimal polynomial over a field extension using Factor(List).

In[7]:= minpoly[a_, e_, x_] :=                                                  
   Select[First/@FactorList[MinimalPolynomial[a, x], Extension->e],             
      PossibleZeroQ[#/.x->a, Method->"ExactAlgebraics"]&][[1]]                  

In[8]:= minpoly[be, al, x]                                                      

              2                      3                           3       2
Out[8]= -1 + x  + x Root[-1 - #1 + #1  & , 1] + Root[-1 - #1 + #1  & , 1]