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I am trying to solve the following problem which I found in a book.

Find a primitive element for the Hilbert class field for $\Bbb{Q}(\sqrt{-17})$? Any hints..

gopal
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    Looking at the factorization of just one integer can yield some hints. For example, $18 = (1 - \sqrt{-17})(1 + \sqrt{-17}) = 2 \times 3^2$, meaning the ring of integers is not a UFD so the class number is greater than $1$, and furthermore, one factorization has one prime factor than the other, so the class number is greater than $2$. –  Jan 05 '15 at 15:57

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According to Franz Lemmermayer's computation the Hilbert class field of $k=\mathbb{Q}(\sqrt{-17})$ is $k(\sqrt{4+\sqrt{17}})=\mathbb{Q}(\sqrt{-17},\sqrt{4+\sqrt{17}})$, see his book Class field towers for details of this computation. The class group $Cl(k)$ is isomorphic to $C_4$, hence the class number of $k$ is equal to $4$. For another example see also here.

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Dietrich Burde
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