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?
How do you prove you've found all possible relations between basis elements? This is especially difficult if you're unlucky in your choice of basis, or in how the field is given. For example, the field $\mathbb{Q}(\sqrt[3]{2},\zeta_3)$ can just as validly be presented as $\mathbb{Q}(\sqrt[3]{4}-5\sqrt[3]{2} + \zeta_3)$. What do you do now?
More importantly, how do you know you've captured all intermediate fields $\mathbb{Q}\subset K\subset \mathbb{Q}(\sqrt[3]{2},\zeta_3)$ that are of the form $K=\mathbb{Q}(a)$ for some $a$ that's not a basis element?