Timeline for Intersection of Hilbert class fields of imaginary quadratic fields

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Sep 20, 2012 at 12:56	comment	added	Filippo Alberto Edoardo		The situation in the case $d\equiv 3\pmod{4}$ is slightly different because the extension $K(i)/K$ needs be ramified at $2$ for the same reason as before. It is then a little bit hard to understand what happens, but if you restrict to $\ell$ being a prime $\equiv 3\pmod{4}$ then $2$ cannot divide the class number of $K$ (see Theorem 10.4 (b) in Washington's Introduction to cyclotomic fields). I know of no standard references, and I would call this kind of results "small/clever/ingenous/trivial trick" according to your taste...Otherwise, Stankewicz' answer correctly pointing to genus theory.
Sep 20, 2012 at 11:37	comment	added	Adam Harris		@Filippo: Thanks! I was thinking about intersections in general though, for example does a similar situation occur for $d \equiv 3 \mod 4$? Is there a standard reference for the statement for $d \equiv 1 \mod 4$?
Sep 20, 2012 at 11:29	history	edited	Filippo Alberto Edoardo	CC BY-SA 3.0	added 690 characters in body
Sep 20, 2012 at 11:17	history	answered	Filippo Alberto Edoardo	CC BY-SA 3.0