eq1 = α == (Sqrt[2 + 3 ( 2^(1/3))])

\(eq2 = Map[#\^2 &, eq1]\)

eq3 = Map[# - 2 &, eq2]

\(eq4 = Map[#\^3 &, eq3]\)

eq5 = Map[# - 54 &, eq4]

minpol = Expand[eq5[[1]] /. α -> x]

Factor[minpol]