Discriminant of an algebraic number field

More specifically, it is proportional to the squared volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified.

The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula for K. A theorem of Hermite states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, and the subject of current research.

The latter is an ideal in the ring of integers of L, and like the absolute discriminant it indicates which primes are ramified in K/L.

The definition of the discriminant of a general algebraic number field, K, was given by Dedekind in 1871.

[22] Near the end of the nineteenth century, Ludwig Stickelberger obtained his theorem on the residue of the discriminant modulo four.

The relative discriminant is defined in a fashion similar to the absolute discriminant, but must take into account that ideals in OL may not be principal and that there may not be an OL basis of OK. Let {σ1, ..., σn} be the set of embeddings of K into C which are the identity on L. If b1, ..., bn is any basis of K over L, let d(b1, ..., bn) be the square of the determinant of the n by n matrix whose (i,j)-entry is σi(bj).

[26] The relative discriminant regulates the ramification data of the field extension K/L.

[29] On the other hand, the existence of an infinite class field tower can give upper bounds on the values of α(ρ, σ).

A fundamental domain of the ring of integers of the field K obtained from Q by adjoining a root of x 3 − x 2 − 2 x + 1. This fundamental domain sits inside K ⊗ Q R . The discriminant of K is 49 = 7 2 . Accordingly, the volume of the fundamental domain is 7 and K is only ramified at 7.
Richard Dedekind showed that every number field possesses an integral basis, allowing him to define the discriminant of an arbitrary number field. [ 16 ]