Determine $n$, $m$ for which $\Bbb Q_n \subseteq \Bbb Q_m$.

Question

Determine $n$, $m$ for which $\Bbb Q_n \subseteq \Bbb Q_m$. How to derive this. I am having no clue. I guess $n|m$. But what is the proof?

Answer 1

Claim $\Bbb Q_n \subseteq \Bbb Q_m$ iff $\phi (m)=\phi (l)$ where $l=l.c.m(m,n)$

$\Leftarrow$

we know $\Bbb Q_n\Bbb Q_m= \Bbb Q_l$. Now $\Bbb Q_m \subseteq \Bbb Q_l$ and $\phi (m)=\phi (l) \Rightarrow [\Bbb Q_l:\Bbb Q]= [\Bbb Q_m:\Bbb Q] \Rightarrow \Bbb Q_l=\Bbb Q_m \Rightarrow \Bbb Q_n \subseteq \Bbb Q_m$

$\Rightarrow$

If $\Bbb Q_n \subseteq \Bbb Q_m \Rightarrow \Bbb Q_n\Bbb Q_m= \Bbb Q_m=\Bbb Q_l \Rightarrow \phi (m)=\phi (l)$