Timeline for Non-cyclotomic abelian extensions

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Nov 19, 2012 at 17:13	comment	added	LMN		Thanks for the link Keith. I think the comments of Lavender Honey and SGP there give a reasonably complete answer. Let P a prime of $\mathbb{Q}$ split in $L$, and consider a quadratic extension $K$ of $L$ ramified at one prime $\mathcal{P}$ over $P$ but not at another $\mathcal{P}'\|P$. Suppose $K \subset L*\mathbb{Q}(\mu_\infty)$, then $K$ is obtained by adjoining some element $\alpha \in \mathbb{Q}(\mu_\infty)$ to $L$. However, looking locally at primes of $L(\alpha)$ over $P$ we see that all the local extensions are the same (infact, they are all $Q_p(\alpha)$)$. But, this is diff. from K\|L
Nov 19, 2012 at 7:08	comment	added	KConrad		Related post: mathoverflow.net/questions/85775/…
Nov 19, 2012 at 5:27	comment	added	LMN		Thanks Chandan! I guess the argument you had in mind is the same as the one below.
Nov 19, 2012 at 5:25	vote	accept	LMN
Nov 19, 2012 at 5:18	answer	added	John Pardon		timeline score: 10
Nov 19, 2012 at 5:08	comment	added	Chandan Singh Dalawat		What prevents $L$ from having an abelian extension which is not even galoisian over $\mathbf{Q}$, let alone abelian over $\mathbf{Q}$ ? Consider for example quadratic extensions of quadratic extensions.
Nov 19, 2012 at 4:48	history	asked	LMN	CC BY-SA 3.0