Linear growth of the number of quintic fields with bounded discriminant

Number theory 1 Minute

You need to know: Prime numbers, infinite product, field, isomorphic and non-isomorphic fields. Also, see this previous theorem description for the concepts of number field, degree of a number field, and discriminant of a number field.

Background: Number fields of degree n=5 are called quintic. For X>0, let N_n(X) denotes the number of non-isomorphic number fields of degree n with absolute value of the discriminant at most X. Also, let {\cal P} be the set of prime numbers.

The Theorem: On 29th September 2004, Manjul Bhargava submitted to the Annals of Mathematics a paper in which he proved that the limit \lim\limits_{X\to\infty}\frac{N_5(X)}{X} exists and is equal to c_5 = \frac{13}{120}\prod\limits_{p\in {\cal P}}\left(1+p^{-2}-p^{-4}-p^{-5}\right) = 0.149....

Short context: Counting number fields up to isomorphism is a basic and important open problem in the area. There is and old folklore conjecture that \lim\limits_{n\to\infty}\frac{N_n(X)}{X} = c_n>0 for every fixed n, but, before 2004, this was known only for n\leq 3 (see here for the best upper bounds for N_n(X) available for general n). In June 2004, Bhargava submitted a paper which proves this conjecture for n=4. The Theorem proves it for n=5 (quintic fields).

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 10.11 of this book for an accessible description of the Theorem.

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Published by Bogdan Grechuk

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