Let K be a finite Galois extension of $\mathbb{Q}$ of degree 12 such that $G\cong \mathbb{Z}_{12}$. I want to determine the number of subfields of $K$ of dimension $n$ over $\mathbb{Q}$ for each $1\leq n \leq 12$.
I am just learn Galois theory so I do not know if I understand correctly, but this is what I have: The subgroups of $Z_{12}$ are $<1>,<2>,<3>,<4>,<6>,<12>=<0>$, so that there are exactly 4 intermediate fields between $K$ and $F$ (is that correct?). So $<2>$ corresponds to a unique subfield of dimension $2$ over $F$, $<3>$ corresponds to a unique subfield of dimension $3$ over $F$, etc.
Is the above correct? I am sorry if this is a bad question. I am new to this and hope to not bother anyone. Thank you for your help.