Hermite–Minkowski theorem

For any integer N there are only finitely many number fields with discriminant at most N

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

| d K | ≥ n n n ! ( π 4 ) n 2 {\displaystyle {\sqrt {|d_{K}|}}\geq {\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

Neukirch, Jürgen (1999). Algebraic Number Theory. Springer. Section III.2