I am looking for a straightforward way to find all intermediate fields of a field extension. Let's take the splitting field of $X^3-2$ over $\Bbb Q$ as an example.
If we adjoin the roots of $X^3-2$ to $\Bbb Q$ we get $\Bbb Q(\sqrt[3]{2}, \zeta_3)$ with $\zeta_3$ a primitive root of unity different from $1$. My idea for finding the intermediate fields would be to look at the $\Bbb Q$-basis derived from the minimal polynomials of the successive extensions $\Bbb Q(\sqrt[3]{2}, \zeta_3)/\Bbb Q(\sqrt[3]{2})/\Bbb Q$.
This basis would be $\{1, \sqrt[3]{2}, (\sqrt[3]{2})^2, \zeta_3, \zeta_3 \sqrt[3]{2}, \zeta_3 (\sqrt[3]{2})^2\}$. Now I look for basis elements which can be turned into other basis elements by field operations like squaring, adding, etc. This is obviously the case for $(\sqrt[3]{2})^2$ but also $(\zeta_3 (\sqrt[3]{2})^2)^2 = -2\zeta_3 \sqrt[3]{2}$. So I get the following intermediate fields: $\Bbb Q(\sqrt[3]{2})$, $\Bbb Q(\zeta_3)$, $\Bbb Q(\zeta_3 \sqrt[3]{2})$.
Does this method give me all intermediate fields? If yes, what's the advantage of an analysis using Galois theory? If no, what am I missing?