Restriction of the Galois group for some abelian extension to some subfield

Question

Assume that $M$ is a finite abelian extension of $\mathbb{Q}$, such that $M\subset \mathbb{Q}(\zeta_m)$ is the smallest cyclotomic field which contains the $M$ via the Kronecker-Weber theorem.

Is the restriction of the Galois group $Gal(\mathbb{Q}(\zeta_m)/\mathbb{Q})$ to subfield $M$ simply the Galois group $Gal(M/\mathbb{Q})$?

Answer 1

We have a tower of normal extensions of $\mathbb Q$: $$ \mathbb Q \subset M \subset \mathbb Q(\zeta_m).$$

In this situation, we have a surjective group homomorphism: $$ \phi : Gal(\mathbb Q(\zeta_m) : \mathbb Q) \to Gal(M : \mathbb Q),$$ which sends each $\sigma \in Gal(\mathbb Q(\zeta_m) : \mathbb Q)$ to its restriction $\sigma|_M \in Gal(M : \mathbb Q)$.

[Since $M : \mathbb Q$ is a normal extension, we have $\sigma(M) = M $, so this definition makes sense. And since $\mathbb Q(\zeta_m) : \mathbb Q$ is a normal extension, every $\mathbb Q$-automorphism of $M$ extends to a $\mathbb Q$-automorphism of $\mathbb Q(\zeta_m)$, which means that $\phi$ is surjective.]

Anyway, the kernel of the group homomorphism $\phi$ is $Gal(\mathbb Q(\zeta_m) : M)$. Hence $\phi$ induces a group isomorphism: $$ \frac{Gal(\mathbb Q(\zeta_m) : \mathbb Q)}{Gal(\mathbb Q(\zeta_m) : M)} \cong Gal(M : \mathbb Q),$$ which is the isomorphism you are looking for.

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