Timeline for Elliptic curves: about a passage in J. Silverman's "Advanced topics of elliptic curves"

Current License: CC BY-SA 4.0

10 events

when toggle format	what		by	license	comment
Jun 27, 2020 at 13:10	history	edited	GH from MO		edited tags
Jun 26, 2020 at 22:51	history	edited	YCor	CC BY-SA 4.0	made title more neutral, formatting
Jun 26, 2020 at 22:11	comment	added	Wojowu		Let $a\in O_K$ be such that $a\in\mathfrak p,a\not\in\mathfrak p^2,a\not\in\mathfrak q$ (CRT). Then $(a)\subseteq\mathfrak p$, so $(a)=\mathfrak a\mathfrak p$ for some $\mathfrak a$. The choice of $a$ guarantees $\mathfrak a$ is not divisible by $\mathfrak p$ or $\mathfrak q$.
Jun 26, 2020 at 20:03	comment	added	Joe Silverman		@NoamD.Elkies I agree, there probably is an easier way. But for a book at this level, I think it's fair to use results from a first course in algbraic number theory. Also, what I wrote isn't quite right, one should take a prime ideal in the ideal class of $\mathfrak p^{-1}$ other than $\mathfrak p$ and $\mathfrak q$.
Jun 26, 2020 at 19:09	comment	added	Noam D. Elkies		It looks even easier because there was no requirement that 𝔞 be prime.
Jun 26, 2020 at 18:40	comment	added	Joe Silverman		Quickest, although maybe overkill, consider the ideal class of $\mathfrak p^{-1}$, Every ideal class contains infinitely many prime ideals, take any of them (other than $\mathfrak p$ itself if $\mathfrak p^2$ is principal) for $\mathfrak a$.
Jun 26, 2020 at 18:23	comment	added	Hair80		Thank you I was not meaning $pO_{K}$ but $\mathfrak{p}$ above $p$, I have edited the question now
Jun 26, 2020 at 18:23	history	edited	Hair80	CC BY-SA 4.0	reference to wrong ideal corrected
Jun 26, 2020 at 18:16	comment	added	Wojowu		Are you sure you've got the question right? $pO_K$ itself is principal, so you could take $\mathfrak a=O_K$.
Jun 26, 2020 at 18:03	history	asked	Hair80	CC BY-SA 4.0