NumberFields@Home: Difference between revisions

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==== 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 .

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 has a bound <math>B</math>, there are finitely many degree <math>n</math> number fields with absolute discriminant less than or equal to <math>B</math>. One can then ask how this count grows as a function of <math>B</math>.

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.

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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.

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 [http://hobbes.la.asu.edu/lowgrd/ 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.

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. [http://wikipedia.org/wiki/P-adic_quantum_mechanics 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 ==

@@ Line 27: / Line 27: @@
 ==== 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 .
+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 has a bound <math>B</math>, there are finitely many degree <math>n</math> number fields with absolute discriminant less than or equal to <math>B</math>. One can then ask how this count grows as a function of <math>B</math>.
 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.
@@ Line 33: / Line 33: @@
 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.
+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 [http://hobbes.la.asu.edu/lowgrd/ 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.
+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. [http://wikipedia.org/wiki/P-adic_quantum_mechanics 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 ==