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.

Definition

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} \ .

Minkowski's constant for the field K is this bound M_K.^[1]

Properties

Since the number of integral ideals of given norm is finite, the finiteness of the class number is an immediate consequence,^[1] and further, the ideal class group is generated by the prime ideals of norm at most M_K.

Minkowski's bound may be used to derive a lower bound for the discriminant of a field K given n, r₁ and r₂. Since an integral ideal has norm at least one, we have 1 ≤ M_K, so that

\sqrt{|D|} \ge \left(\frac{\pi}{4}\right)^{r_2} \frac{n^n}{n!} \ge \left(\frac{\pi}{4}\right)^{n/2} \frac{n^n}{n!} \ .

For n at least 2, it is easy to show that the lower bound is greater than 1, so we obtain Minkowski's Theorem, that the discriminant of every number field, other than Q, is non-trivial. This implies that the field of rational numbers has no unramified extension.

Proof

The result is a consequence of Minkowski's theorem.

References

1 2 Pohst & Zassenhaus (1989) p.384

Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. 62 (2nd printing of 1st ed.). Springer-Verlag. ISBN 3-540-63003-1. Zbl 0819.11044.
Lang, Serge (1994). Algebraic Number Theory. Graduate Texts in Mathematics 110 (second ed.). New York: Springer. ISBN 0-387-94225-4. Zbl 0811.11001.
Pohst, M.; Zassenhaus, H. (1989). Algorithmic Algebraic Number Theory. Encyclopedia of Mathematics and its Applications 30. Cambridge University Press. ISBN 0-521-33060-2. Zbl 0685.12001.

External links

Using Minkowski's Constant To Find A Class Number at PlanetMath.org.
Stevenhagen, Peter. Number Rings.
The Minkowski Bound at Secret Blogging Seminar

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