In mathematics, especially in

algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

, the Hermite–Minkowski theorem states that for any integer ''N'' there are only finitely many

number fields In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a fi ...

, i.e., finite

field extension In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...

s ''K'' of the rational numbers Q, such that the

discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coef ...

of ''K''/Q is at most ''N''. The theorem is named after

Charles Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermite p ...

and

Hermann Minkowski Hermann Minkowski (22 June 1864 – 12 January 1909) was a mathematician and professor at the University of Königsberg, the University of Zürich, and the University of Göttingen, described variously as German, Polish, Lithuanian-German, o ...

. 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 In mathematics, Stirling's approximation (or Stirling's formula) is an Asymptotic analysis, asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirli ...

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