Hermite–Minkowski theorem

Template:Short description 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

$\sqrt{|d_K|}\ge \frac{n^n}{n!}{\left(\frac{\pi }{4}\right)}^{\frac{n}{2}}$

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

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