NumberFields@Home

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

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 categorize 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. 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 categorize fields is by their discriminant, which is an important invariant for a field. Given a fixed bound , there are only a finite number of fields whose discriminant is less than this bound. A secondary goal of the project is to determine the finite set of "minimum discriminant" imprimitive decic fields for the bound $B=1.2\; times\; 10^\{11\}$. This bound was chosen for it's potential to find more fields while keeping the computational load manageable.

Methods

Computing lower degree fields requires less processing power and have been more extensively tabulated but the degree 10 case is the first case requiring a massively parallel solution. After recalling this article in Linux Magazine, and knowing that the math department had limited resources (but access to a suitable workstation), Eric Driver launched NumberFields@Home to meet the computational demand.

Finite extension fields are represented by polynomials (i.e. they are of the form $mathbb\{Q\}(alpha)$ where $alpha$ 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 we are searching for. At the most basic level, the NumberFields@Home algorithm searches over this finite set of polynomials, checking whether or not a polynomial can represent a field with the desired discriminant and ramification properties. At a finer level, the algorithm uses some tricky 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 dissertation.

The project as a whole is basic research, in effect, charting unknown territory. In the future, this may have a bearing on a number of questions.

Automorphic Forms

Number fields are related to automorphic forms, which are part of the Langlands program. Explanations for the Langlands program. (See video: The Biggest Project in Modern Mathematics) is an automorphic form in the complex plane.]]The theory of automorphic forms is an important topic within mathematics. They provide one side of the Langlands program, a set of sweeping conjectures in number theory. There are deep connections between automorphic forms and number fields, and knowing an automorphic form will give information about the ramifying primes of corresponding number fields.

Cryptography

Number fields are used in some modern factoring algorithms which are relevant to attacks on RSA. Other researchers have investigated using properties of number fields as the basis of new cryptosystems. It is not clear what number fields will be 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 $n$ and has a bound $B$, there are finitely many degree $n$ number fields with absolute discriminant less than or equal to $B$. One can then ask how this count grows as a function of $B$.

Recently, researchers have been factoring the Galois group of the extension. 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. This project has helped establish the first examples of imprimitive decic number fields with certain Galois groups. One can also consider "first examples" from another perspective, namely by the Galois root discriminant (GRD) of the field. We compute the GRD of the fields found there, looking for fields with especially small GRD. Some results for low GRD fields can be found here.

Theoretical Physics

The fields concerned with in 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 our tables of fields will be to the physics community.

Project team / Sponsors

Eric D. Driver. school of mathematics at Arizona State University.

Scientific results

* https://numberfields.asu.edu/NumberFields/FieldTables/FieldTables.html

Scientific publications

# Driver, Eric D. and John W. Jones. Computing septic number fields. Journal of Number Theory (2019). DOI: 10.1016/j.jnt.2019.02.022.

# Driver, Eric D. and John W. Jones. Minimum Discriminants of Imprimitive Decic Fields. Experimental Mathematics (2010). DOI: 10.1080/10586458.2010.10390637.

# Driver, Eric D. and John W. Jones. A targeted Martinet search. Mathematics of Computation (2009). DOI: 10.1090/S0025-5718-08-02178-9.

Contributing

If you're interested in supporting this project, download and install BOINC and attach to the project using its official URL: https://numberfields.asu.edu/NumberFields/.