NumberFields@Home

NumberFields@Home
NumberFields@Home logo
Project
Status	Active
Category	Mathematics / Algebraic Number Theory
Compute	CPU & GPU
Requires	None
Development
Developer	Eric D. Driver
Author	Eric D. Driver
Sponsor	Arizona State University School of Mathematics
Maintainer	Eric D. Driver, Greg Tucker
Initial release	August 2011
Software
Written in	C, C++ (PARI/GP, GMP)
Operating system	Windows, Linux, macOS (x86)
BOINC statistics
Stats as of	May 2026
Performance	~47,566 GigaFLOPS
Metadata
Website	https://numberfields.asu.edu/NumberFields/

BOINC project NumberFields@Home is a volunteer distributed computing project that needs your help to search for fields with special properties.

A visualisation of the algebraic numbers in the complex plane. Number fields are algebraic extensions of the rationals whose elements lie among these points.

Background

Fields are important mathematical constructs with far-reaching applications across many branches of mathematics. Most people are familiar with everyday examples such as the field of rational numbers <math>\mathbf{Q}</math>, the real numbers <math>\mathbf{R}</math>, and the complex numbers <math>\mathbf{C}</math>. The fields studied by NumberFields@Home are number fields: algebraic extension fields of the rationals of finite degree. More precisely, a number field is a subset of <math>\mathbf{C}</math> which contains the root <math>\alpha</math> of a given polynomial and is minimal while remaining closed under addition, subtraction, multiplication, and division (excepting division by zero).

Formally, every number field <math>K</math> of degree <math>n</math> over <math>\mathbf{Q}</math> may be written as

<math>K = \mathbf{Q}(\alpha)</math>

where <math>\alpha</math> is a root of some irreducible polynomial of degree <math>n</math> with rational coefficients. The project is principally interested in imprimitive degree-10 fields (called decic fields), which correspond to certain degree-10 polynomials. An imprimitive field is one that contains a proper intermediate subfield strictly between <math>\mathbf{Q}</math> and itself, as opposed to primitive fields whose Galois closure has an irreducible Galois group.

Number theorists can mine tabulated data for patterns to help formulate conjectures about number fields, leading to a deeper understanding of the properties of numbers, the basic building blocks of all mathematics. Among the practical applications are cryptography, where number fields underpin sophisticated factoring algorithms and novel cryptosystems, and theoretical physics including quantum mechanics and string theory.

History

The project was founded by Eric D. Driver, a researcher associated with the School of Mathematics at Arizona State University (ASU). Driver recognised that computing lower-degree number fields requires comparatively modest resources and that such fields had already been extensively tabulated, but the degree-10 case was the first case demanding a massively parallel computational solution. After reading an article in Linux Magazine about BOINC and knowing the mathematics department had access to a suitable workstation, he launched NumberFields@Home to meet the computational demand.

The project went online in August 2011. In an early forum post, Driver noted that the project had been under construction and work-generation processes still needed automation - yet the response from the distributed computing community was enthusiastic enough to quickly exhaust the initial work queue.

A significant milestone arrived in May 2016, when the primary bounded application completed its multi-year search and found all imprimitive degree-10 fields with absolute discriminant less than or equal to <math>1.2 \times 10^{11}</math>. In July 2016, a focused special search located a particularly elusive hypothesised field: an <math>A_5</math> extension of <math>\mathbf{Q}(\sqrt{421})</math> ramified only at 2, after roughly ten months of intermittent searching.

In August 2022, Driver shared news that his doctoral thesis advisor - the project's primary institutional benefactor at ASU - had retired. While the university permitted the project to continue running, it would no longer fund hardware upgrades, meaning the project's lifespan is now tied to the longevity of the existing server hardware. As of 2026, NumberFields@Home remains active, collaborating with the BOINC-based Gerasim@Home project to cross-check and complete certain sub-searches.

Why NumberFields@Home?

Fields are important mathematical constructs that have far-reaching applications to many branches of mathematics. Many people are familiar with the fields of rational numbers, real numbers, and complex numbers. The fields we are concerned with in this project are number fields: subsets of the complex numbers which contain the root of a given polynomial and are minimal for then being closed under addition, subtraction, multiplication, and division (except for division by 0). In particular, we are interested in imprimitive degree 10 fields (called decic fields), which correspond to certain degree 10 polynomials.

Number theorists can mine the data for interesting patterns to help them formulate conjectures about number fields. Ultimately, this research will lead to a deeper understanding of the properties of numbers, the basic building blocks of all mathematics. Another application of number fields is in cryptography, where they are used in sophisticated factoring algorithms and as the basis for new cryptosystems. There are also distant applications to mathematical physics, including quantum mechanics and string theory.

Goals

One way to categorise fields is by the primes that ramify in them. For a given set of primes, the number of fields ramified at those primes is finite (a classical theorem of Hermite). The primary goal of the project is to find this finite set of fields for various sets of primes. Since the number of combinations of primes is unlimited, the project will remain open-ended for the foreseeable future.

Another way to categorise fields is by their discriminant, which is an important invariant. Given a fixed bound <math>B</math>, there are only finitely many fields whose absolute discriminant does not exceed <math>B</math>. A secondary goal of the project is to determine the finite set of minimum discriminant imprimitive decic fields for the bound

<math>B = 1.2 \times 10^{11}</math>

This bound was chosen for its potential to find more fields while keeping the computational load manageable. That search was completed in May 2016.

The graph of a degree-3 polynomial. NumberFields@Home searches over polynomials of degree 10 whose roots generate number fields with prescribed properties.

Methods

Computing lower-degree fields requires less processing power and such fields have been more extensively tabulated; the degree-10 case is the first case requiring a massively parallel solution. Finite extension fields are represented by polynomials - that is, they are of the form <math>\mathbf{Q}(\alpha)</math>, where <math>\alpha</math> is the root of a polynomial. Bounds on the field discriminant give rise to bounds on the polynomial coefficients, so there are a finite number of possible polynomials that can represent the fields being searched for.

At the most basic level, the NumberFields@Home algorithm searches over this finite set of polynomials, checking whether or not a given polynomial can represent a field with the desired discriminant and ramification properties. At a finer level, the algorithm uses theoretical arguments to reduce the polynomial search space. In addition, the targeted ramification structure gives rise to congruence relations on the polynomial coefficients, which further reduces the search space. Anybody interested in the finer details of the algorithm is encouraged to look through Eric D. Driver's doctoral dissertation.

Software stack

The application relies on two key open-source libraries:

• PARI/GP - a computer algebra system widely used in number theory, providing polynomial arithmetic and discriminant computations.
• GMP (GNU Multiple Precision Arithmetic Library) - multi-precision integer arithmetic, necessary because the integers involved can exceed standard 64-bit representation.

A significant technical challenge arose during GPU application development: both PARI/GP and GMP rely on dynamically allocated memory, which is incompatible with GPU kernels. Driver solved this by using a fixed-precision multi-precision library with precision hard-coded to the maximum required (approximately 750 bits), allowing compilation of a working GPU kernel.

Application versions

The project currently distributes its Get Decic Fields application in multiple variants targeting different hardware:

Platform	Variant	Notes
Windows x86-64	Default (CPU) v4.00	~20,622 GigaFLOPS average
Windows x86-64	OpenCL AMD	GPU variant
Windows x86-64	OpenCL NVIDIA	GPU variant
Linux x86-64	Default (CPU) v4.00	~8,860 GigaFLOPS average
Linux x86-64	CUDA	NVIDIA GPU variant
Linux x86-64	OpenCL AMD	AMD GPU variant
macOS (Intel)	Default (CPU) v3.00	~887 GigaFLOPS average

The GPU application is reported to be roughly 20 to 30 times faster than the older CPU version, and 2 to 3 times faster than a newer optimised CPU version released alongside the GPU work. The combined average computing across all platforms is approximately 47,566 GigaFLOPS.

Results database

The results of NumberFields@Home contribute to an online searchable number field database maintained jointly by John W. Jones and David P. Roberts at ASU. Tabulated results are also integrated into the LMFDB (L-functions and Modular Forms Database).

Research areas

The project as a whole is basic research - in effect, charting unknown mathematical territory. The following sections describe the broader research contexts to which the tabulated data contributes.

Automorphic Forms

Number fields are related to automorphic forms, which form one side of the Langlands program - a vast web of conjectures connecting number theory and representation theory. Explanations for the Langlands program. (See video: The Biggest Project in Modern Mathematics)

The Dedekind eta-function is an automorphic form in the complex plane.

There are deep connections between automorphic forms and number fields: knowing an automorphic form gives information about the ramifying primes of the corresponding number fields.

Cryptography

Number fields are used in some modern factoring algorithms relevant to attacks on RSA. Other researchers have investigated using properties of number fields as the basis for new cryptosystems. It is not clear which number fields will prove most useful in this research, but the more we know about the general landscape of number fields, the better.

Arithmetic Statistics

There has been both progress and new conjectures in recent years on asymptotic questions about number fields. If one fixes the degree <math>n</math> and a bound <math>B</math>, there are finitely many degree-n number fields with absolute discriminant at most <math>B</math>. One can then ask how this count grows as a function of <math>B</math>.

Researchers have also been factoring the Galois group of the extension into consideration. At present, there is very little data in degree 10, and imprimitive fields produce a large number of different Galois groups.

One can also ask about asymptotics based on the set of ramifying primes. There is even less data currently available for investigating questions of this sort. Before one can seriously consider asymptotics, it is useful to know where the first examples lie. NumberFields@Home has helped establish the first known examples of imprimitive decic number fields with certain Galois groups. One can also consider first examples from the perspective of the Galois root discriminant (GRD) of the field. Some results for low GRD fields can be found here.

Theoretical Physics

The fields studied by this project have connections to the p-adic fields. In recent years, p-adic analysis has been applied to problems in theoretical physics, including quantum mechanics and string theory. Here is a good introduction to the relevant concepts. It is too early to tell exactly how beneficial the tables of fields will be to the physics community.

Project team / Sponsors

• Eric D. Driver - Project founder, administrator, developer, and scientist. School of Mathematics at Arizona State University.
• Greg Tucker - assists Driver with the project and its applications.

The project is based at and was sponsored by the ASU School of Mathematics. Following the retirement of Driver's thesis advisor in May 2022, the primary institutional sponsorship ended, though ASU has permitted the project to continue on existing hardware.

Scientific results

• Complete tables of imprimitive decic number fields with absolute discriminant <math>\leq 1.2 \times 10^{11}</math> (completed May 2016).
• First known examples of imprimitive decic fields for numerous Galois group types.
• Tables of low Galois root discriminant (GRD) imprimitive decic fields: results for low GRD fields.
• Full results in table form: https://numberfields.asu.edu/NumberFields/FieldTables/FieldTables.html
• Searchable database integration with the LMFDB: www.lmfdb.org/NumberField/

Scientific publications

1. Driver, Eric D. and John W. Jones. Computing septic number fields. Journal of Number Theory (2019). DOI: 10.1016/j.jnt.2019.02.022.
2. Driver, Eric D. and John W. Jones. Minimum Discriminants of Imprimitive Decic Fields. Experimental Mathematics (2010). DOI: 10.1080/10586458.2010.10390637.
3. Driver, Eric D. and John W. Jones. A targeted Martinet search. Mathematics of Computation (2009). DOI: 10.1090/S0025-5718-08-02178-9.
4. Jones, John W. and David P. Roberts. A database of number fields. LMS Journal of Computation and Mathematics 17 (2014), pp. 595-618. DOI: 10.1112/S1461157014000424.

See also

• Berkeley Open Infrastructure for Network Computing (BOINC)
• Algebraic number field
• Discriminant of an algebraic number field
• Langlands program
• Volunteer computing

External links

• NumberFields@Home official website
• ASU Number Fields Database
• LMFDB Number Fields search
• Eric D. Driver's doctoral dissertation (PDF)
• Low Galois root discriminant results
• Eric D. Driver - A Number Fields Database (PARI workshop slides, 2016)