# Computing the Hilbert class field

Does anyone know any good source with nice examples of Hilbert class field computations? I'm trying to piece together the theory with some canonical examples.

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You will find a lot of material on quadratic, cubic and quartic unramified extensions of quadratic number fields in the Seminar on complex multiplication (Borel, Chowla, Herz, Iwasawa, Serre) published in Springer's Lecture Notes 21. Beware of typos, however.

The primary source for unramified cyclic extensions of number fields is the thesis by G. Gras, parts of which were published in Extensions abéliennes non ramifiées de degré premier d'un corps quadratique, Bull. Soc. Math. Fr. 100 (1972), 177-193; there are, however, no exercises there.

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Marcus's Number Fields has a bunch of exercises on computing Hilbert Class fields of quadratic fields at the end of Chapter 8, but they are exercises. E.g., exercise 17 asks for the determination of the Hilbert class field and the Hilbert$^+$ class field for $\mathbb{Q}[\sqrt{m}]$ for $2\leq m\leq 10$, $m$ square free. Exercise 24 asks to show that the Hilbert class field of $\mathbb{Q}[\sqrt{-23}]$ is obtained by adjoining a root of $x^3-x+1$; that of $\mathbb{Q}[\sqrt{-31}]$ by adjoining a root of $x^3+x+1$; and that of $\mathbb{Q}[\sqrt{-139}]$ by adjoining a root of $x^3-8x+9$. But these are exercises, rather than worked out examples, so they may not be what you are looking for.

Cohen's A Course in Computational Algebraic Number Theory has an algorithm for computing the Hilbert Class polynomial of quadratic imaginary number fields (Section 7.6.2, algorithm 7.6.1), so that may be closer to what you want, but for a rather restricted class.

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Thanks, I'll work through those. Do you know of any cubic examples? – dstt Mar 28 '11 at 16:30
@dstt: No; and you might want to wait on accepting the answer until you see if one of our resident number theorists can give you a better one! – Arturo Magidin Mar 28 '11 at 16:36
Alright, I'll wait on accepting it. – dstt Mar 28 '11 at 16:46
@M Turgeon: You bumped a question that didn't have activity for over a year, just the replace Hilbert${}^{+}$ with Hilbert$^+$? It does not even render differently; what's the point? – Arturo Magidin Jul 18 '12 at 23:37