NumberFields@home

NumberFields@home is a volunteer computing project that searches for algebraic number fields with specific properties — one of the central objects of study in algebraic number theory. A number field is an extension of the rational numbers obtained by adjoining roots of polynomial equations; for example, adjoining the square root of 2 to the rationals creates the simplest non-trivial number field. The study of number fields is fundamental to modern mathematics, connecting algebra, geometry, and analysis in profound ways.

Based at Arizona State University and led by Professor John Jones, the project systematically enumerates number fields of given degree and bounded discriminant. The discriminant of a number field is an integer that encodes essential arithmetic information about the field — its ramification behavior, the structure of its ring of integers, and its relationship to other fields. Finding all number fields with discriminant below a given bound is a computationally intensive task that grows exponentially with the degree of the field.

The project uses Hunter's theorem and related bounds to organize the search space, then tests candidate polynomials for irreducibility and computes the discriminants and Galois groups of the resulting fields. Volunteers' computers evaluate millions of candidate polynomials, and successful finds are added to the project's publicly accessible database — one of the most comprehensive number field tables in existence, used by algebraic number theorists worldwide.

This database serves as an essential resource for testing conjectures about the distribution of number fields, class numbers, regulators, and other arithmetic invariants. The data has been used in research published in leading mathematics journals and helps advance our understanding of some of the deepest structures in mathematics, including connections to the Langlands program — one of the grand unifying visions of modern mathematics.