Discriminant of An Algebraic Number Field - Basic Results

Basic Results

  • Brill's Theorem: The sign of the discriminant is (−1)r2 where r2 is the number of complex places of K.
  • A prime p ramifies in K if, and only if, p divides ΔK .
  • Stickelberger's Theorem:
  • Minkowski's bound: Let n denote the degree of the extension K/Q and r2 the number of complex places of K, then
  • Minkowski's Theorem: If K is not Q, then |ΔK| > 1 (this follows directly from the Minkowski bound).
  • Hermite's Theorem: Let N be a positive integer. There are only finitely many algebraic number fields K with |ΔK| < N.

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