Timeline for An interesting unramified extension of imaginary quadratic fields

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Nov 6 at 15:54	comment	added	Aurel		After a second look I realise your extension is actually $K(\sqrt{(-1)^{(p-1)/2}p})/K$, since $\mathop{Frob}_{\mathfrak{q}} \mapsto (N(\mathfrak{q}) \bmod p)$ is the mod $p$ cyclotomic character. In particular it is definitely ramified at the primes above $p$ under your hypotheses.
Nov 3 at 17:09	comment	added	Tam Nguyen		@Aurel thank you very much, you're right. Someone else also pointed out this mistake. I wasn't in my right mind to think that the inert primes are all of the principal ones. And indeed it has to be ramified somewhere, in which case it has to be ramified at $p$ if I'm not mistaken.
Nov 3 at 9:39	comment	added	Aurel		I don't understand your proof that $L$ is contained in $H_K$. You need to prove that every principal prime ideal of $K$ splits in $L$, but you checked it only for inert primes (a trivial way of being principal). You extension smells ramified at $p$.
Nov 3 at 0:48	history	asked	Tam Nguyen	CC BY-SA 4.0