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Suppose $K=Q(\alpha)$, $\alpha$ is a root of $X^3-17X+31$,prove that the class number of $K$ is $1$. The Minkowski number is too large, I have no ideal of that. Who can help me?

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up vote 4 down vote accepted

The Minkowski bound is only 22.4, so you just need to factorise each of the 8 primes $p < 22$ in K and check whether the factors are principal. That's not exactly a long calculation. (It might help you to know that $\mathbb{Z}[\alpha]$ is the full ring of integers of $K$.)

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I can just do the case that the Minkowski bound is less than 4,but here,maybe you can show the situation of (3) as a basic example. –  Strongart Aug 28 '11 at 6:04
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