$\mathbb{Q}({\mu }_m)\cap \mathbb{Q}({\mu }_n)=\mathbb{Q}({\mu }_{\gcd (m,n)})$

Given: Integers $m,n\ge 1$ with $d=\gcd (m,n)$ and $\mathrm{\ell }=\mathrm{lcm}(m,n)$; $\mathbb{Q}$ is an ordered field (The rationals form a totally ordered field), so $\mathrm{char}\mathbb{Q}=0$ (The characteristic of a ring: the least $n\ge 1$ with $n\cdot 1_R=0$ when one exists, and $0$ otherwise) and divides no positive integer (Divisibility in $\mathbb{Z}$: $d\mid a$ when $a=dq$ for some integer $q$); a splitting field $\mathrm{\Omega }$ of $t^{\mathrm{\ell }}-1$ over $\mathbb{Q}$; and, for each positive divisor $k$ of $\mathrm{\ell }$, the subfield $\mathbb{Q}({\mu }_k):=\mathbb{Q}({\mu }_k(\mathrm{\Omega }))$ of $\mathrm{\Omega }$. Write $I:=\mathbb{Q}({\mu }_m)\cap \mathbb{Q}({\mu }_n)$.

For a positive divisor $k$ of $\mathrm{\ell }$, the subfield $\mathbb{Q}({\mu }_k(\mathrm{\Omega }))$ generated by the $k$-th roots of unity in $\mathrm{\Omega }$ is a cyclotomic extension of $\mathbb{Q}$ of order $k$ (The cyclotomic extension $K({\mu }_n)$ as a splitting field of $t^n-1$, The group ${\mu }_n(K)$ of $n$-th roots of unity in a field, and primitive $n$-th roots of unity); because $\mathrm{char}\mathbb{Q}=0$ divides no positive integer, $t^n-1$ is separable over $K$ exactly when the characteristic does not divide $n$, and then a splitting field carries $n$ distinct $n$-th roots of unity gives ${\mu }_k(\mathrm{\Omega })$ cyclic of order $k$, with exactly $\phi (k)$ primitive $k$-th roots of unity.

$[\mathbb{Q}({\mu }_k):\mathbb{Q}]=\phi (k)$ for every $k\ge 1$ ($[\mathbb{Q}({\zeta }_n):\mathbb{Q}]=\phi (n)$ and $\mathrm{Gal}(\mathbb{Q}({\mu }_n)\mathrm{/}\mathbb{Q})\cong (\mathbb{Z}\mathrm{/}n{)}^{\times }$, The unit group $(\mathbb{Z}\mathrm{/}n{)}^{\times }$ and Euler's totient $\phi (n)=\mid (\mathbb{Z}\mathrm{/}n{)}^{\times }\mid$ for $n\ge 1$, The degree $[K:F]={\dim }_{F}K$ of a finite field extension).

$\mathbb{Q}({\mu }_k)\mathrm{/}\mathbb{Q}$ is finite Galois ($K({\mu }_n)\mathrm{/}K$ is Galois and $\sigma \mapsto a_{\sigma }$ embeds its Galois group into $(\mathbb{Z}\mathrm{/}n{)}^{\times }$).

For $E\mathrm{/}F$ finite Galois and $L\mathrm{/}F$ finite inside a common field, $[EL:F]=[E:F][L:F]\mathrm{/}[E\cap L:F]$ (For $E\mathrm{/}F$ finite Galois and $L\mathrm{/}F$ finite inside a common field, $[EL:F]=[E:F][L:F]\mathrm{/}[E\cap L:F]$).

$\mathbb{Q}({\mu }_m)\mathbb{Q}({\mu }_n)=\mathbb{Q}({\mu }_{\mathrm{\ell }})$ inside $\mathrm{\Omega }$ ($K({\mu }_m)K({\mu }_n)=K({\mu }_{\mathrm{lcm}(m,n)})$).

$\phi (m)\phi (n)=\phi (d)\phi (\mathrm{\ell })$ ($\phi (m)\phi (n)=\phi (\gcd (m,n))\phi (\mathrm{lcm}(m,n))$).

For fields $F\subseteq K\subseteq M$ with $K\mathrm{/}F$ and $M\mathrm{/}K$ finite, $[M:F]=[M:K][K:F]$ (Tower law for finite extensions: $[L:F]=[L:K][K:F]$).