I am trying to find the proper subfields between $\mathbb Q(2^{1/3},\omega)$ and $\mathbb Q$ . where $\omega$ is a cube root of unity . I have found that finding the galois groups and the proper subfields one has to be very careful , and its very easy to conclude something wrong .
Here taking this particular case , finding the possible automorphism which keep $\mathbb Q$ fixed we observe the following things ,
The vector space of $\mathbb Q(2^{1/3},\omega)$ over $\mathbb Q $ will have elements of the form ,
$a+b 2^{1/3}+c 2^{2/3} +d\omega +e 2^{1/3} \omega +f 2^{2/3} \omega$ .
I find it quite problematic to find whether mapping one "root " to the other gives an automorphism or not .
Is it true that first we need to find the set of linearly independent "basis" of the extension before we find the set of automorphism .
May be my question is very vague but my question is what are the very basic facts one needs to take care while finding galois group and the proper subfields.