Discriminant of An Algebraic Number Field - Relation To Other Quantities

Relation To Other Quantities

• When embedded into, the volume of the fundamental domain of O_K is (sometimes a different measure is used and the volume obtained is, where r₂ is the number of complex places of K).
• Due to its appearance in this volume, the discriminant also appears in the functional equation of the Dedekind zeta function of K, and hence in the analytic class number formula, and the Brauer–Siegel theorem.
• The relative discriminant of K/L is the Artin conductor of the regular representation of the Galois group of K/L. This provides a relation to the Artin conductors of the characters of the Galois group of K/L, called the conductor-discriminant formula.