Hermite–Minkowski Theorem

(Theorems In Algebraic Number Theory)

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Theorems In Algebraic Number Theory
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What is 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 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 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.

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theorems in algebraic number theory

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Hermite-Minkowski theorem

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