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



Find a cubic monic integral coefficient whose discriminant is $-83$
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.


