Minkowski's bound

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Minkowski's bound describes an upper limit for computing the class number of a field $K$ over $\mathbb{Q}$ . It is named for the mathematician Hermann Minkowski. Let $D$ be the discriminant of the field, $n$ be the degree of $K$ over $\mathbb{Q}$ , and $2 r 2 = n - r 1$ be the number of complex embeddings where $r 1$ is the number of real embeddings. Then every class in the ideal class group of $K$ contains an integral ideal of norm not exceeding Minkowski's bound

$M_K = \sqrt{|D|} \left(\frac{4}{\pi}\right)^{r_2} \frac{n!}{n^n}.$

In particular, the class group is generated by the prime ideals of norm at most $M K$ .

[edit] References

Using Minkowski's Constant To Find A Class Number on PlanetMath
Lang, Serge (1994). Algebraic Number Theory, second edition, New York: Springer. ISBN 0387942254.
Stevenhagen, Peter. Number Rings.

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