Discriminant of an algebraic number field
![A fundamental domain of the ring of integers of the field K obtained from Q by adjoining a root of x3 − x2 − 2x + 1. This fundamental domain sits inside K ⊗QR. The discriminant of K is 49 = 72. Accordingly, the volume of the fundamental domain is 7 and K is only ramified at 7.](/Images/godic/202502/06/Discriminant49CubicFieldFundamentalDomain4605.png")
![Richard Dedekind showed that every number field possesses an integral basis, allowing him to define the discriminant of an arbitrary number field.[15]](/Images/godic/202502/06/Dedekind4605.jpeg")
In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified.