Discriminant of an algebraic number field

The discriminant of an algebraic number field is a numerical invariant in algebraic number theory that measures the size of the field's ring of integers; geometrically, it is proportional to the squared volume of a fundamental domain of that ring of integers. It is computed as the square of a determinant built from the embeddings of the number field into the complex numbers together with an integral basis of the ring of integers. The discriminant carries structural information about the field, since the prime numbers that divide it are exactly the primes that ramify in the ring of integers, and it appears in central analytic formulas, including the functional equation of the field's Dedekind zeta function and the analytic class number formula. The Hermite-Minkowski theorem shows that only finitely many number fields can share a given discriminant, although determining that exact number for a given discriminant remains an open problem in general. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/