Timeline for Hilbert class field of Quadratic fields

8 events

when toggle format	what		by	license	comment
Jan 23, 2021 at 14:30	history	edited	GH from MO		edited tags
Jan 23, 2021 at 14:19	answer	added	Davood Khajehpour		timeline score: 1
Sep 28, 2011 at 12:49	comment	added	Cam McLeman		For your second question, the answer is no by genus theory: The more primes that divide d, the bigger the class group.
Sep 28, 2011 at 12:37	comment	added	Tommaso Centeleghe		Observe that the ring of Integers of $Q(\sqrt{-d})$ when $d\equiv 1$ mod $4$ is not a UFD because the only ideal lying above $2$, which is ramified, cannot be principal (the integer $2$ is not a norm). The theory of complex multiplication (in case you did not know) relates the Hilbert class field of an imaginary quadratic filed K to the j-invariant of any elliptic curve with CM by the integers of K (one can learn this in "Primes of the form x^2+ny^2" by D.cox)
Sep 28, 2011 at 12:21	history	edited	Sina	CC BY-SA 3.0	added 13 characters in body
Sep 28, 2011 at 12:18	comment	added	Emerton		It is not true that $\mathbb Q(\sqrt{d},i)$ is the Hilbert Class Field of $\mathbb Q(\sqrt{-d})$ in general when $4 \| (d-1)$, just that it is contained in the Hilbert Class Field.
Sep 28, 2011 at 11:39	comment	added	Dror Speiser		A book dedicated to this question is Cohen's Advanced Topics in Computational Number Theory.
Sep 28, 2011 at 11:12	history	asked	Sina	CC BY-SA 3.0