ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 30 May 2017 15:28:02 -0500Can I efficiently verify if given $h$ is the class number of a quadratic field?http://ask.sagemath.org/question/8422/can-i-efficiently-verify-if-given-h-is-the-class-number-of-a-quadratic-field/Can I efficiently verify if given $h$ is the class number of a quadratic field?
Computing the class number is not tractable.
I tried pari's `Qfb` and did some experiments with Lidia, but I must be missing something.
What I tried is for random $a$ compute $a^h$ but I don't get the identity and in Lidia I can't check if it is principal.Sun, 30 Oct 2011 01:26:49 -0500http://ask.sagemath.org/question/8422/can-i-efficiently-verify-if-given-h-is-the-class-number-of-a-quadratic-field/Comment by dan_fulea for <p>Can I efficiently verify if given $h$ is the class number of a quadratic field?</p>
<p>Computing the class number is not tractable.</p>
<p>I tried pari's <code>Qfb</code> and did some experiments with Lidia, but I must be missing something.</p>
<p>What I tried is for random $a$ compute $a^h$ but I don't get the identity and in Lidia I can't check if it is principal.</p>
http://ask.sagemath.org/question/8422/can-i-efficiently-verify-if-given-h-is-the-class-number-of-a-quadratic-field/?comment=37743#post-id-37743If i correctly understand the question, the question is as follows: There are given a quadratic number field $K=\mathbb Q(\sqrt D)$ and a to-be-class number integer $h^?$. We want to verify $h^?=h(K)$ efficiently, in particular without calling the computation of the class number.
(So the question is NOT if for a given $h$ there exists a $K$ realizing $h=h(K)$...)
Is it enough / necessary to verify, that $h^?$ is a multiple of $h(K)$? If yes, then one may have "only" to compute $a$ to the power $h^?$ for each $a$ in the Minkowski cage insured by theory. But we are still not able to claim $h^?=h(K)$, but only $h(K)\ |\ h^?$. This may not even be really effective for the half problem.
The question is a sage question or a math one?
If it is a math question, why not use analytic methods?Tue, 30 May 2017 15:28:02 -0500http://ask.sagemath.org/question/8422/can-i-efficiently-verify-if-given-h-is-the-class-number-of-a-quadratic-field/?comment=37743#post-id-37743