Minkowski's bound

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 = nr1 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 MK.

The result is a consequence of Minkowski's theorem.

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