I am attempting to compute the minimal polynomial of $1 + 2^{1/3} + 4^{1/3}$ over $\mathbb Q$. So far, my reasoning is as follows:
The Galois conjugates of $2^{1/3}$ are $2^{1/3} e^{2\pi i/3}$ and $2^{1/3} e^{4\pi i /3}$. We have $4^{1/3} = 2^{2/3}$, so the image of $4^{1/3}$ under an automorphism $\sigma$ fixing $\mathbb Q$ is determined by the image of $2^{1/3}$: it must equal the square of $\sigma(2^{1/3})$. Therefore, the Galois conjugates of $1 + 2^{1/3} + 4^{1/3}$ are $1 + 2^{1/3} e^{2\pi i/3} + 4^{1/3} e^{4\pi i/3}$ and $1 + 2^{1/3} e^{4\pi i/3} + 4^{1/3} e^{2\pi i/3}$. Therefore, the minimal polynomial is $(x-a)(x-b)(x-c)$, where
$\begin{align*} a&=1 + 2^{1/3} + 4^{1/3},\\ b&=1 + 2^{1/3} e^{2\pi i/3} + 4^{1/3} e^{4\pi i/3},\text{ and}\\ c&=1 + 2^{1/3} e^{4\pi i/3} + 4^{1/3} e^{2\pi i/3}. \end{align*}$
However, this polynomial does not seem to have coefficients in $\mathbb Q$! What am I doing wrong?