MinimalPolynomial[s, x, Extension->a] finds the characteristic polynomial of the element s of Q[a] over the field 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]

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]