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If we want to bound the norm of the smallest ideal which generates a nontrivial ideal class, is there a better bound than Minkowski's bound?

(Note that Minkowski's bound is to guarantee something much stronger, namely that every ideal class has a representative whose norm is less than the given bound. Here, we just want that some nontrivial ideal class has such a representative.)

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3 Answers 3

The classical reference to an improvement of Minkowski bounds is

  • R. Zimmert, Ideale kleiner Norm in Idealklassen und eine Regulatorabschätzung, Invent. Math. 62 (1980), 367-380
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If you are willing to assume the GRH (of course, once you have found your ideal, you can verify it unconditionally), then there is Bach's bound: E. Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), 355-380. It is also concerned with bounds on generators of the entire ideal class group (so might still be worse than some other bound for your weaker requirement), but is much better than Minkowski's bound. You might also be interested in this article of Karim Belabas and in the references therein.

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Improving substantially on Minkowski's bound is a difficult problem. I recommend you Section 6 of Einsiedler-Lindenstrauss-Michel-Venkatesh: Distribution of periodic torus orbits on homogeneous spaces, Duke Math. J. 148 (2009), 119-174. Here it is shown that, in a certain sense, improvement is possible for most number fields and also for all number fields with a large regulator.

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