How can I prove that $\mathbb{Q}(\sqrt[3]{2},i)=\mathbb{Q}(\sqrt[3]{2}+i)$? It is easy to show that $\mathbb{Q}(\sqrt[3]{2}+i)\subseteq \mathbb{Q}(\sqrt[3]{2},i)$. 
But how I can show that $\mathbb{Q}(\sqrt[3]{2},i)\subseteq\mathbb{Q}(\sqrt[3]{2}+i)$?
I can't find a way to express $\sqrt[3]{2}$ in terms of $\sqrt[3]{2}+i$.

If you are lazy and do not want to compute by hand, or you want to check a greater number of similar examples and be sure there is no slip of the pen in the solution, you can do it with Macaulay2. Note:
1) iJ is generated by the irreducible polynomial, that $z = i + 2^{1/3}$ has to fulfill.
2) The primaryDecomposition calculations below give the representations of $v = p(z)$ for $v = i$ ($v^2 + 1 = 0$) and $v = 2^{1/3}$ ($v^3 - 2 = 0$). You can read them off from the factors of the primary decomposition. If you have several possibilities (several factors contain an $p_i(z) = v$, here the case at $v^2+1=0$) then you must embed everything into (a floating point approximation) of the complex (or real) numbers and calculate which $p_i(z) = v$ is fulfilled. Increase precision, until only one $p_{i_0}(z) = v$ is floating point zero.
Macaulay2, version 1.12
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems, LLLBases,
               PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : R=QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : iI = ideal(x^2+1, y^3-2)

             2       3
o2 = ideal (x  + 1, y  - 2)

o2 : Ideal of R

i3 : R1=R/iI

o3 = R1

o3 : QuotientRing

i4 : S=QQ[z]

o4 = S

o4 : PolynomialRing

i5 : phi=map(R1,S,{x+y})

o5 = map(R1,S,{x + y})

o5 : RingMap R1 <--- S

i6 : iJ = ker phi

            6     4     3     2
o6 = ideal(z  + 3z  - 4z  + 3z  + 12z + 5)

o6 : Ideal of S

i7 : S1 = S/iJ

o7 = S1

o7 : QuotientRing

i8 : S11 = S1[v]

o8 = S11

o8 : PolynomialRing

i9 : primaryDecomposition ideal (S11/ideal(v^2 + 1))

              2               5     4      3      2                      2               5     4  
o9 = {ideal (v  + 1, 22v + 12z  - 9z  + 40z  - 78z  + 78z + 91), ideal (v  + 1, 22v - 12z  + 9z  -
     -----------------------------------------------------------------------------------------------
        3      2
     40z  + 78z  - 78z - 91)}

o9 : List

i10 : primaryDecomposition ideal (S11/ideal(v^3-2))

               3               5     4      3      2                       3         2       5     4
o10 = {ideal (v  - 2, 22v - 12z  + 9z  - 40z  + 78z  - 100z - 91), ideal (v  - 2, 22v  + (12z  - 9z 
      ----------------------------------------------------------------------------------------------
           3      2                    5     4      3       2
      + 40z  - 78z  + 100z + 91)v - 18z  + 8z  - 60z  + 106z  - 106z - 142)}

o10 : List

