Minkowski's bound

In algebraic number theory, Minkowski's bound gives an upper bound of the norm of ideals to be checked in order to determine the class number of a number field K. 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 2r₂ = n − r₁ be the number of complex embeddings where r₁ 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.

The result is a consequence of Minkowski's theorem.

References

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

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