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
\( {\sqrt {|d_{K}|}}\geq {\frac {n^{n}}{n!}}\left({\frac \pi 4}\right)^{{n/2}} \)
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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