Hermite–Minkowski theorem

In mathematics, especially in algebraic number theory, the Hermite–Minkowski theorem states that for any integer ''N'' there are only finitely many number fields, i.e., finite field extensions ''K'' of the rational numbers Q, such that the discriminant of ''K''/Q is at most ''N''. The theorem is named after Charles Hermite and Hermann Minkowski.

This theorem is a consequence of the estimate for the discriminant

: $\sqrt \geq \frac\left(\frac\pi4\right)^$

where ''n'' is the degree of the field extension, together with Stirling's formula for ''n''!. This inequality also shows that the discriminant of any number field strictly bigger than Q is not ±1, which in turn implies that Q has no unramified extensions.

References

Section III.2

{{DEFAULTSORT:Hermite-Minkowski theorem
Theorems in algebraic number theory