i have a question on two infinite extensions of $\mathbb{Q}$. Write $\mathbb{Q}^{ab}$ for the maximal abelian extension of $\mathbb{Q}$, that is
$$\mathbb{Q}^{ab} = \prod_{L/\mathbb{Q} \mbox{ abelian}}L$$
and let $\mathbb{Q}(\mu_{\infty})$ be the maximal cyclotomic extension of $\mathbb{Q}$, that is, $\mathbb{Q}$ together with all the roots of unity.
Is it true that these two fields are the same? Namely, do we have $\mathbb{Q}^{ab} = \mathbb{Q}(\mu_{\infty})$?
My attempt:
It is clear that $\mathbb{Q}(\mu_{\infty}) \subset \mathbb{Q}^{ab}$, as the first field can be seen as an inverse limit of cyclotomic (hence abelian) finite extensions.
My guess for the other inclusion is that we can write
$$\mathbb{Q}^{ab} = \prod_{\substack{L/\mathbb{Q} \mbox{ abelian}\\ \mbox{finite}}}L$$
so that for any $L/\mathbb{Q}$ finite abelian, we use Kronecker-Weber's theorem and embed $L$ into a cyclotomic extension of $\mathbb{Q}$. In this way $\mathbb{Q}^{ab}$ would be included in $\mathbb{Q}(\mu_{\infty})$. However, i am not sure if this argument is correct, and i am not even sure if these two fields are equal or not.
Thanks in advance!
You are right: these two fields are equal, and your proof is fine.
You should replace "an inverse limit of cyclotomic (hence abelian) finite extensions" by "a direct limit [...]". The use of Kronecker--Weber theorem is essential here. This doesn't work directly for general number fields, see here (although there is the theory of complex multiplication or more generally class field theory to describe abelian extensions of some number fields).