NumberFields@Home
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NumberFields@Home is a volunteer distributed computing project that needs your help to ...
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.
Goal
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 <math>B=1.2 \times 10^{11}</math>. We chose this bound 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. The degree 10 case is the first case requiring a massively parallel solution, and hence the reason for implementing a BOINC project.
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