This post continues my series on favorite theorems of the twenty-first century. For an overview of the categories and my earlier selections, see this introductory post.
My choice for 2004 in Algebra is Manjul Bhargava’s paper The Density of Discriminants of Quintic Rings and Fields. The paper was received by the Annals of Mathematics in 2004 and appeared in print in 2010 (Bhargava 2010).
A number field is a finite extension of . Its dimension as a vector space over is called its degree and is denoted by
For every , multiplication by defines a -linear map The trace of this linear transformation is called the field trace of and is denoted by Since the trace of a linear transformation is independent of the choice of basis, this definition is intrinsic to the field extension .
An element is called an algebraic integer if it is a root of a monic polynomial with integer coefficients. The algebraic integers in form a ring, denoted by and called the ring of integers of . As an abelian group, is free of rank . A basis of as a -module is called an integral basis of .
The discriminant of is This is a nonzero integer that does not depend on the choice of integral basis.
For , let denote the number of isomorphism classes of number fields of degree satisfying The Hermite–Minkowski theorem implies that is finite for every fixed and . An old folklore conjecture predicts that, for every fixed , there is a positive constant such that Equivalently, the conjecture asserts that
The case is elementary. In 1971, Davenport and Heilbronn (Davenport and Heilbronn 1971) proved the conjecture for cubic fields. In 2005, Bhargava (Bhargava 2005) established the next case, and proved that the number of quartic number fields satisfies where
n the paper selected here, Bhargava proved the conjecture for quintic fields and obtained an explicit formula for the asymptotic constant.
The proofs for quartic and quintic fileds follow the same broad strategy. Bhargava’s remarkable algebraic parametrizations encode quartic and quintic rings as integral orbits in concrete representation spaces. This transforms the problem of counting number fields into a problem of counting lattice points of bounded discriminant. Methods from the geometry of numbers provide the main asymptotic count, while a delicate sieve removes the nonmaximal orders and leaves precisely the rings of integers of number fields.
The quartic argument relies on Bhargava’s parametrization of quartic rings, while the quintic argument uses his parametrization of quintic rings, discussed in this earlier post.