View source for 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 extension]]s ''K'' of the rational numbers '''Q''', such that the [[discriminant of an algebraic number field|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

: <math>\sqrt{|d_K|} \geq \frac{n^n}{n!}\left(\frac\pi4\right)^{n/2}</math>

where ''n'' is the degree of the field extension, together with [[Stirling's formula]] for ''n''<nowiki>!</nowiki>. 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 [[Splitting of prime ideals in Galois extensions|unramified]] extensions.

==References==
{{cite book|last=Neukirch|first=Jürgen|title=Algebraic Number Theory|year=1999|publisher=Springer}} Section III.2

{{DEFAULTSORT:Hermite-Minkowski theorem}}
[[Category:Theorems in algebraic number theory]]