I trie to calculate the Hilbert class field of $\mathbb Q\left(\sqrt{-83}\right)$, whose class number is $3$, so I should find a cubic integral monic polynomial whose discriminant is $-83$, but I failed
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$\begingroup$ Please use TeX on this site. $\endgroup$– GH from MOCommented Mar 17, 2020 at 19:34
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There's no guarantee that such a polynomial exists, because the corresponding cubic subfield of the Hilbert class field need not be monogenic. But it does actually exist in this case and one example is
$$x^3 - x^2 + x - 2$$
A really useful website for looking up polynomials whose corresponding number field has small discriminant and/or constrained ramification is this one:
https://hobbes.la.asu.edu/NFDB/
You can find the polynomial above by setting the degree to be $3$ and the the number of ramified primes to be 1 with smallest and largest prime $83$.
You can of course find the same field using Kummer theory but it would be a slow process.
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$\begingroup$ What ? See en.wikipedia.org/wiki/Primitive_element_theorem $\endgroup$– F. C.Commented Mar 17, 2020 at 21:04
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3$\begingroup$ @F.C. are you objecting to "because the corresponding cubic subfield of the Hilbert class field need not be monogenic"? The primitive element theorem refers to the generator of a field; monogenic refers to the generator of the ring of integers; no contradiction there. $\endgroup$ Commented Mar 17, 2020 at 22:57
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$\begingroup$ This tool is really useful,thanks a lot ! $\endgroup$ Commented Mar 18, 2020 at 7:41
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$\begingroup$ @F.C. What ? See en.wikipedia.org/wiki/Mansplaining $\endgroup$ Commented Mar 20, 2020 at 12:40