Discriminant of an algebraic number field

In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the 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 prime#In [algebraic number theory|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, and the class number formula">class number (number theory)">class number formula for. 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 discriminant of can be referred to as the absolute discriminant of to distinguish it from the relative discriminant of an extension of number fields. The latter is an ideal in the ring of integers of, and like the absolute discriminant it indicates which primes are ramified in. It is a generalization of the absolute discriminant allowing for to be bigger than ; in fact, when, the relative discriminant of is the principal ideal of generated by the absolute discriminant of.

Definition

Let be an algebraic number field, and let be its ring of integers. Let be an integral basis of, and let be the set of embeddings of into the complex numbers. The discriminant of is the square of the determinant of the matrix whose -entry is. Symbolically,
Equivalently, the trace from to can be used. Specifically, define the trace form to be the matrix whose -entry is. This matrix equals, so defining the discriminant of as the determinant of this matrix gives us the same result as the preceding definition.
The discriminant of an order in with integral basis is defined in the same way.

Examples

Quadratic number fields: let be a square-free integer; then the discriminant of is
Cyclotomic fields: let be an integer, let be a [Root of unity|primitive n-th root of unity], and let be the -th cyclotomic field. The discriminant of is given by
Power bases: In the case where the ring of algebraic integers has a power integral basis, that is, can be written as, the discriminant of is equal to the discriminant of the Minimal [polynomial (field theory)|minimal polynomial] of. To see this, one can choose the integral basis of to be
Let be the number field obtained by adjoining a root of the polynomial. This is Richard Dedekind's original example of a number field whose ring of integers does not possess a power basis. An integral basis is given by
Repeated discriminants: the discriminant of a quadratic field uniquely identifies it, but this is not true, in general, for higher-degree number fields. For example, there are two non-isomorphic cubic fields of discriminant 3969. They are obtained by adjoining a root of the polynomial or, respectively.

Basic results

Brill's theorem: The sign of the discriminant is where is the number of complex places of.
A prime ramifies in if and only if divides.
Stickelberger's theorem:
Minkowski's bound: Let denote the degree of the extension and the number of complex places of, then
Minkowski's theorem: If is not, then .
Hermite–Minkowski theorem: Let be a positive integer. There are only finitely many algebraic number fields with. Again, this follows from the Minkowski bound together with Hermite's theorem.

History

The definition of the discriminant of a general algebraic number field, K, was given by Dedekind in 1871. At this point, he already knew the relationship between the discriminant and ramification.
Hermite's theorem predates the general definition of the discriminant with Charles Hermite publishing a proof of it in 1857. In 1877, Alexander von Brill determined the sign of the discriminant. Leopold Kronecker first stated Minkowski's theorem in 1882, though the first proof was given by Hermann Minkowski in 1891. In the same year, Minkowski published his bound on the discriminant. Near the end of the nineteenth century, Ludwig Stickelberger obtained his theorem on the residue of the discriminant modulo four.

Relative discriminant

The discriminant defined above is sometimes referred to as the absolute discriminant of K to distinguish it from the relative discriminant Δ_K/''L of an extension of number fields K''/L, which is an ideal in O_L. The relative discriminant is defined in a fashion similar to the absolute discriminant, but must take into account that ideals in O_L may not be principal and that there may not be an O_L basis of O_K. Let be the set of embeddings of K into C which are the identity on L. If b₁,..., b_n is any basis of K over L, let d be the square of the determinant of the n by n matrix whose -entry is σ_i. Then, the relative discriminant of K/''L is the ideal generated by the d'' as varies over all integral bases of K/''L. Alternatively, the relative discriminant of K''/L is the norm of the different of K/''L. When L'' = Q, the relative discriminant Δ_K/Q is the principal ideal of Z generated by the absolute discriminant Δ_K . In a tower of fields K/''L/F'' the relative discriminants are related by
where denotes relative norm.

Ramification

The relative discriminant regulates the ramification data of the field extension K/''L. A prime ideal p'' of L ramifies in K if, and only if, it divides the relative discriminant Δ_K/''L''. An extension is unramified if, and only if, the discriminant is the unit ideal. The Minkowski bound above shows that there are no non-trivial unramified extensions of Q. Fields larger than Q may have unramified extensions: for example, for any field with class number greater than one, its Hilbert class field is a non-trivial unramified extension.

Root discriminant

The root discriminant of a degree n number field K is defined by the formula
The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension.

Asymptotic lower bounds

Given nonnegative rational numbers ρ and σ, not both 0, and a positive integer n such that the pair = is in Z × 2Z, let α_n be the infimum of rd_K as K ranges over degree n number fields with r real embeddings and 2s complex embeddings, and let α = liminf_n→∞α_n. Then
and the generalized Riemann hypothesis implies the stronger bound
There is also a lower bound that holds in all degrees, not just asymptotically: For totally real fields, the root discriminant is > 14, with 1229 exceptions.

Asymptotic upper bounds

On the other hand, the existence of an infinite class field tower can give upper bounds on the values of α. For example, the infinite class field tower over Q with m = 3·5·7·11·19 produces fields of arbitrarily large degree with root discriminant 2 ≈ 296.276, so α < 296.276. Using tamely ramified towers, Hajir and Maire have shown that α < 954.3 and α < 82.2, improving upon earlier bounds of Martinet.

Relation to other quantities

When embedded into, the volume of the fundamental domain of O_K is .
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.