NumberFields@Home: Difference between revisions

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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 [https://www.linux-magazine.com/Issues/2006/71/BOINC/ '''''article in Linux Magazine'''''], and knowing that the math department had limited resources (but access to a suitable workstation), a BOINC project was launched.

Finite extension fields are represented by polynomials (i.e. they are of the form where 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 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 [https://numberfields.asu.edu/NumberFields/Dissertation.pdf Eric D. Driver's dissertation].

Number fields are related to [[wikipedia:Automorphic_form|'''''automorphic forms''''']], which are part of the Langlands program. '''[https://www.quantamagazine.org/what-is-the-langlands-program-20220601/ Explanations for the Langlands program]'''''. ('''''[https://youtu.be/_bJeKUosqoY See video: The Biggest Project in Modern Mathematics]''''')''

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 [[wikipedia:Automorphic_form|'''''automorphic forms''''']], which are part of the Langlands program. '''[https://www.quantamagazine.org/what-is-the-langlands-program-20220601/ Explanations for the Langlands program]'''''. ('''''[https://youtu.be/_bJeKUosqoY See video: The Biggest Project in Modern Mathematics]''''')''[[Image:Dedekind Eta.jpg|none|thumb|500px|The [[wikipedia:Dedekind_eta_function|'''''Dedekind eta-function''''']] 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 and has a bound , there are finitely many degree number fields with absolute discriminant less than or equal to . One can then ask how this count grows as a function of .

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.

== Project team / Sponsors ==

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 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 [https://www.linux-magazine.com/Issues/2006/71/BOINC/ '''''article in Linux Magazine'''''], and knowing that the math department had limited resources (but access to a suitable workstation), a BOINC project was launched.
-[[Image:Dedekind Eta.jpg|none|thumb|500px|The [[wikipedia:Dedekind_eta_function|'''''Dedekind eta-function''''']] is an automorphic form in the complex plane.]]
+Finite extension fields are represented by polynomials (i.e. they are of the form  where  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 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 [https://numberfields.asu.edu/NumberFields/Dissertation.pdf Eric D. Driver's dissertation].
-Number fields are related to [[wikipedia:Automorphic_form|'''''automorphic forms''''']], which are part of the Langlands program. '''[https://www.quantamagazine.org/what-is-the-langlands-program-20220601/ Explanations for the Langlands program]'''''.  ('''''[https://youtu.be/_bJeKUosqoY See video: The Biggest Project in Modern Mathematics]''''')''
+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 [[wikipedia:Automorphic_form|'''''automorphic forms''''']], which are part of the Langlands program. '''[https://www.quantamagazine.org/what-is-the-langlands-program-20220601/ Explanations for the Langlands program]'''''.  ('''''[https://youtu.be/_bJeKUosqoY See video: The Biggest Project in Modern Mathematics]''''')''[[Image:Dedekind Eta.jpg|none|thumb|500px|The [[wikipedia:Dedekind_eta_function|'''''Dedekind eta-function''''']] 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  and has a bound , there are finitely many degree  number fields with absolute discriminant less than or equal to . One can then ask how this count grows as a function of .
+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.
 == Project team / Sponsors ==