\( \def\bbZ{ {\mathbb Z} } \def\bbQ{ {\mathbb Q} } \def\bbR{ {\mathbb R} } \def\bbC{ {\mathbb C} } \def\st{\hspace{8pt} \middle| \hspace{8pt}} \def\Gal{\operatorname{Gal}} \)
NumberFields@home Project Description

Project Goals

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. Computing lower degree fields requires less processing power and have been more extensively tabulated. The degree 10 case is the first case requiring a massively parallel solution, and hence the reason for implementing a BOINC project.

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 forseeable future.

Another way to categorize fields is by their discriminant, which is an important invariant for a field. Given a fixed bound \(B\), 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 \( B=1.2 \times 10^{11} \). We chose this bound for it's potential to find more fields while keeping the computational load manageable.


Applications of the Project

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

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.

For example, in studying automorphic forms over \( \bbQ(i) \), a researcher wanted to show that there were no automorphic forms of a certain type. The problem was recast into a statement about number fields, summarized by the following statement:

There is no quintic extension \( L \) of \( \bbQ(i) \) satisfying:
  1. \( \Gal( L^g/\bbQ(i))=A_5 \) (where \(L^g\) is the Galois closure of \(L\) ),
  2. \( L \) is unramified outside of \( S=\{2,5\} \), and
  3. \( d_L \) divides \( 2^{14}5^{15} \).
Using the table of decic fields over \( \bbQ(i) \), this statement was shown to be true. (Note that the project is still completing the search over \( \bbQ(i) \), but the discriminant bound above was small enough that the earlier part of the search was sufficient to determine the truth of the statement.)

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 here, 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.


Algorithmic Details

Finite extension fields are represented by polynomials (i.e. they are of the form \( \bbQ(\alpha) \) 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 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 targetted 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 my dissertation.