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), Eric Driver launched NumberFields@Home to meet the computational demand.

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

Finite extension fields are represented by polynomials (i.e. they are of the form ℚ<math>(\alpha)</math> where <math>\alpha</math> 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 [https://numberfields.asu.edu/NumberFields/Dissertation.pdf '''''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.

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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), Eric Driver launched NumberFields@Home to meet the computational demand.
-Finite extension fields are represented by polynomials (i.e. they are of the form <math>(\alpha)</math> 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 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 [https://numberfields.asu.edu/NumberFields/Dissertation.pdf '''''Eric D. Driver's dissertation'''''].
+Finite extension fields are represented by polynomials (i.e. they are of the form ℚ<math>(\alpha)</math> where <math>\alpha</math> 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 [https://numberfields.asu.edu/NumberFields/Dissertation.pdf '''''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.