Minkowski's bound

In

Let D be the discriminant of the field, n be the degree of K over ${\displaystyle \mathbb {Q} }$, and ${\displaystyle 2r_{2}=n-r_{1}}$ be the number of

${\displaystyle r_{1}}$

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

${\displaystyle {\sqrt {|D|}}\geq \left({\frac {\pi }{4}}\right)^{r_{2}}{\frac {n^{n}}{n!}}\geq \left({\frac {\pi }{4}}\right)^{n/2}{\frac {n^{n}}{n!}}\ .}$

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

Proof

The result is a consequence of Minkowski's theorem.

References

• Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. Vol. 62 (2nd printing of 1st ed.).
  Zbl 0819.11044
  .
• Zbl 0811.11001
  .
• Pohst, M.;
  Zbl 0685.12001
  .

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