Al Piskun at 11:54, 31 May 2026

2026-05-31T11:54:06Z

← Older revision		Revision as of 11:54, 31 May 2026
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	{{Infobox software		{{Infobox software
	\| name = NumberFields@Home		\| name = NumberFields@Home
	\| logo = ~~NumberFields logo~~.~~png~~		\| logo = Nf.jpg
	\| logo caption = NumberFields@Home logo		\| logo caption = NumberFields@Home logo
	\| status = Active		\| status = Active

Al Piskun: re-add references

2026-05-31T11:44:41Z

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Al Piskun at 11:46, 30 May 2026

2026-05-30T11:46:25Z

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Admin: update and infobox

2026-05-30T11:35:40Z

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Al Piskun at 11:24, 30 May 2026

2026-05-30T11:24:35Z

← Older revision		Revision as of 11:24, 30 May 2026
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	\| stats as of = May 2026		\| stats as of = May 2026
	\| average performance = ~47,566 GigaFLOPS		\| average performance = ~47,566 GigaFLOPS
	\| website = numberfields.asu.edu/NumberFields/		\| website = https://numberfields.asu.edu/NumberFields/
	}}		}}

Admin at 11:21, 30 May 2026

2026-05-30T11:21:49Z

← Older revision		Revision as of 11:21, 30 May 2026
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	}}		}}

	BOINC project '''''NumberFields@home''''' is a '''''[[wikipedia:Volunteer computing\|volunteer distributed computing]]''''' project that needs your help to search for fields with special properties.		BOINC project '''''[https://numberfields.asu.edu/NumberFields/ NumberFields@home]''''' is a '''''[[wikipedia:Volunteer computing\|volunteer distributed computing]]''''' project that needs your help to search for fields with special properties.

	[[Image:Algebraicszoom.png\|right\|thumb\|300px\|A visualisation of the [[wikipedia:Algebraic_number\|algebraic numbers]] in the [[wikipedia:Complex_plane\|complex plane]]. Number fields are algebraic extensions of the rationals whose elements lie among these points.]]		[[Image:Algebraicszoom.png\|right\|thumb\|300px\|A visualisation of the [[wikipedia:Algebraic_number\|algebraic numbers]] in the [[wikipedia:Complex_plane\|complex plane]]. Number fields are algebraic extensions of the rationals whose elements lie among these points.]]

Al Piskun at 03:01, 30 May 2026

2026-05-30T03:01:59Z

← Older revision		Revision as of 03:01, 30 May 2026
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	{{Infobox software		{{Infobox software
	\| name = NumberFields@~~Home~~		\| name = NumberFields@home
	\| logo = ~~NumberFields logo~~.~~png~~		\| logo = Nf.jpg
	\| logo caption = NumberFields@~~Home~~ logo		\| logo caption = NumberFields@home logo
	\| status = Active		\| status = Active
	\| category = Mathematics / Algebraic Number Theory		\| category = Mathematics / Algebraic Number Theory
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	}}		}}

	BOINC project '''''NumberFields@~~Home~~''''' is a '''''[[wikipedia:Volunteer computing\|volunteer distributed computing]]''''' project that needs your help to search for fields with special properties.		BOINC project '''''NumberFields@home''''' is a '''''[[wikipedia:Volunteer computing\|volunteer distributed computing]]''''' project that needs your help to search for fields with special properties.

	[[Image:Algebraicszoom.png\|right\|thumb\|300px\|A visualisation of the [[wikipedia:Algebraic_number\|algebraic numbers]] in the [[wikipedia:Complex_plane\|complex plane]]. Number fields are algebraic extensions of the rationals whose elements lie among these points.]]		[[Image:Algebraicszoom.png\|right\|thumb\|300px\|A visualisation of the [[wikipedia:Algebraic_number\|algebraic numbers]] in the [[wikipedia:Complex_plane\|complex plane]]. Number fields are algebraic extensions of the rationals whose elements lie among these points.]]
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	== Background ==		== Background ==

	Fields are important mathematical constructs with far-reaching applications across many branches of mathematics. Most people are familiar with everyday examples such as the field of [[wikipedia:Rational_number\|rational numbers]] <math>\mathbf{Q}</math>, the [[wikipedia:Real_number\|real numbers]] <math>\mathbf{R}</math>, and the [[wikipedia:Complex_number\|complex numbers]] <math>\mathbf{C}</math>. The fields studied by NumberFields@~~Home~~ are '''[[wikipedia:Algebraic_number_field\|number fields]]''': algebraic extension fields of the rationals of finite degree. More precisely, a number field is a subset of <math>\mathbf{C}</math> which contains the root <math>\alpha</math> of a given polynomial and is minimal while remaining closed under addition, subtraction, multiplication, and division (excepting division by zero).{{cite web \|title=What is NumberFields@home?\|publisher=Arizona State University \|access-date=2026-05-29}}		Fields are important mathematical constructs with far-reaching applications across many branches of mathematics. Most people are familiar with everyday examples such as the field of [[wikipedia:Rational_number\|rational numbers]] <math>\mathbf{Q}</math>, the [[wikipedia:Real_number\|real numbers]] <math>\mathbf{R}</math>, and the [[wikipedia:Complex_number\|complex numbers]] <math>\mathbf{C}</math>. The fields studied by NumberFields@home are '''[[wikipedia:Algebraic_number_field\|number fields]]''': algebraic extension fields of the rationals of finite degree. More precisely, a number field is a subset of <math>\mathbf{C}</math> which contains the root <math>\alpha</math> of a given polynomial and is minimal while remaining closed under addition, subtraction, multiplication, and division (excepting division by zero).{{cite web \|title=What is NumberFields@home?\|publisher=Arizona State University \|access-date=2026-05-29}}

	Formally, every number field <math>K</math> of degree <math>n</math> over <math>\mathbf{Q}</math> may be written as		Formally, every number field <math>K</math> of degree <math>n</math> over <math>\mathbf{Q}</math> may be written as
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	== History ==		== History ==

	The project was founded by '''Eric D. Driver''', a researcher associated with the School of Mathematics at '''[[wikipedia:Arizona_State_University\|Arizona State University]]''' (ASU). Driver recognised that computing lower-degree number fields requires comparatively modest resources and that such fields had already been extensively tabulated, but the degree-10 case was the first case demanding a massively parallel computational solution. After reading an article in ''Linux Magazine'' about BOINC and knowing the mathematics department had access to a suitable workstation, he launched NumberFields@~~Home~~ to meet the computational demand.{{cite web \|title=NumberFields@home - Methods\|publisher=Arizona State University \|access-date=2026-05-29}}		The project was founded by '''Eric D. Driver''', a researcher associated with the School of Mathematics at '''[[wikipedia:Arizona_State_University\|Arizona State University]]''' (ASU). Driver recognised that computing lower-degree number fields requires comparatively modest resources and that such fields had already been extensively tabulated, but the degree-10 case was the first case demanding a massively parallel computational solution. After reading an article in ''Linux Magazine'' about BOINC and knowing the mathematics department had access to a suitable workstation, he launched NumberFields@home to meet the computational demand.{{cite web \|title=NumberFields@home - Methods\|publisher=Arizona State University \|access-date=2026-05-29}}

	The project went online in '''August 2011'''. In an early forum post, Driver noted that the project had been under construction and work-generation processes still needed automation - yet the response from the distributed computing community was enthusiastic enough to quickly exhaust the initial work queue.{{cite web \|title=NumberFields@home still under construction\|publisher=numberfields.asu.edu \|date=8 August 2011 \|access-date=2026-05-29}}		The project went online in '''August 2011'''. In an early forum post, Driver noted that the project had been under construction and work-generation processes still needed automation - yet the response from the distributed computing community was enthusiastic enough to quickly exhaust the initial work queue.{{cite web \|title=NumberFields@home still under construction\|publisher=numberfields.asu.edu \|date=8 August 2011 \|access-date=2026-05-29}}
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	A significant milestone arrived in '''May 2016''', when the primary ''bounded'' application completed its multi-year search and found all imprimitive degree-10 fields with absolute discriminant less than or equal to <math>1.2 \times 10^{11}</math>. In '''July 2016''', a focused special search located a particularly elusive hypothesised field: an <math>A_5</math> extension of <math>\mathbf{Q}(\sqrt{421})</math> ramified only at 2, after roughly ten months of intermittent searching.{{cite web \|title=2016 year in review\|publisher=numberfields.asu.edu \|date=14 January 2017 \|access-date=2026-05-29}}		A significant milestone arrived in '''May 2016''', when the primary ''bounded'' application completed its multi-year search and found all imprimitive degree-10 fields with absolute discriminant less than or equal to <math>1.2 \times 10^{11}</math>. In '''July 2016''', a focused special search located a particularly elusive hypothesised field: an <math>A_5</math> extension of <math>\mathbf{Q}(\sqrt{421})</math> ramified only at 2, after roughly ten months of intermittent searching.{{cite web \|title=2016 year in review\|publisher=numberfields.asu.edu \|date=14 January 2017 \|access-date=2026-05-29}}

	In '''August 2022''', Driver shared news that his doctoral thesis advisor - the project's primary institutional benefactor at ASU - had retired. While the university permitted the project to continue running, it would no longer fund hardware upgrades, meaning the project's lifespan is now tied to the longevity of the existing server hardware.{{cite web \|title=Future of the Project\|publisher=numberfields.asu.edu \|date=10 August 2022 \|access-date=2026-05-29}} As of 2026, NumberFields@~~Home~~ remains active, collaborating with the '''[[wikipedia:BOINC\|BOINC]]'''-based Gerasim@Home project to cross-check and complete certain sub-searches.{{cite web \|title=2025 Year End Summary\|publisher=numberfields.asu.edu \|date=1 January 2026 \|access-date=2026-05-29}}		In '''August 2022''', Driver shared news that his doctoral thesis advisor - the project's primary institutional benefactor at ASU - had retired. While the university permitted the project to continue running, it would no longer fund hardware upgrades, meaning the project's lifespan is now tied to the longevity of the existing server hardware.{{cite web \|title=Future of the Project\|publisher=numberfields.asu.edu \|date=10 August 2022 \|access-date=2026-05-29}} As of 2026, NumberFields@home remains active, collaborating with the '''[[wikipedia:BOINC\|BOINC]]'''-based Gerasim@Home project to cross-check and complete certain sub-searches.{{cite web \|title=2025 Year End Summary\|publisher=numberfields.asu.edu \|date=1 January 2026 \|access-date=2026-05-29}}

	== Why NumberFields@~~Home~~? ==		== 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.		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.
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	Computing lower-degree fields requires less processing power and such fields have been more extensively tabulated; the degree-10 case is the first case requiring a massively parallel solution. Finite extension fields are represented by polynomials - that is, they are of the form <math>\mathbf{Q}(\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 being searched for.		Computing lower-degree fields requires less processing power and such fields have been more extensively tabulated; the degree-10 case is the first case requiring a massively parallel solution. Finite extension fields are represented by polynomials - that is, they are of the form <math>\mathbf{Q}(\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 being searched for.

	At the most basic level, the NumberFields@~~Home~~ algorithm searches over this finite set of polynomials, checking whether or not a given polynomial can represent a field with the desired discriminant and ramification properties. At a finer level, the algorithm uses 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 [numberfields.asu.edu/NumberFields/Dissertation.pdf Eric D. Driver's doctoral dissertation].{{cite web \|title=Eric D. Driver's dissertation\|publisher=Arizona State University \|access-date=2026-05-29}}		At the most basic level, the NumberFields@home algorithm searches over this finite set of polynomials, checking whether or not a given polynomial can represent a field with the desired discriminant and ramification properties. At a finer level, the algorithm uses 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 [numberfields.asu.edu/NumberFields/Dissertation.pdf Eric D. Driver's doctoral dissertation].{{cite web \|title=Eric D. Driver's dissertation\|publisher=Arizona State University \|access-date=2026-05-29}}

	=== Software stack ===		=== Software stack ===
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	=== Results database ===		=== Results database ===

	The results of NumberFields@~~Home~~ contribute to an online searchable number field database maintained jointly by John W. Jones and David P. Roberts at ASU. Tabulated results are also integrated into the '''[[wikipedia:L-functions_and_Modular_Forms_Database\|LMFDB]]''' (L-functions and Modular Forms Database).{{cite web \|title=NumberFields@home\|publisher=Arizona State University \|access-date=2026-05-29}}{{cite journal \|last1=Jones \|first1=John W. \|last2=Roberts \|first2=David P. \|title=A database of number fields \|journal=LMS Journal of Computation and Mathematics \|year=2014 \|volume=17 \|pages=595-618 \|doi=10.1112/S1461157014000424}}		The results of NumberFields@home contribute to an online searchable number field database maintained jointly by John W. Jones and David P. Roberts at ASU. Tabulated results are also integrated into the '''[[wikipedia:L-functions_and_Modular_Forms_Database\|LMFDB]]''' (L-functions and Modular Forms Database).{{cite web \|title=NumberFields@home\|publisher=Arizona State University \|access-date=2026-05-29}}{{cite journal \|last1=Jones \|first1=John W. \|last2=Roberts \|first2=David P. \|title=A database of number fields \|journal=LMS Journal of Computation and Mathematics \|year=2014 \|volume=17 \|pages=595-618 \|doi=10.1112/S1461157014000424}}

	== Research areas ==		== Research areas ==
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	Researchers have also been factoring the [[wikipedia:Galois_group\|Galois group]] of the extension into consideration. At present, there is very little data in degree 10, and imprimitive fields produce a large number of different Galois groups.		Researchers have also been factoring the [[wikipedia:Galois_group\|Galois group]] of the extension into consideration. 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. NumberFields@~~Home~~ has helped establish the first known examples of imprimitive decic number fields with certain Galois groups. One can also consider ''first examples'' from the perspective of the '''Galois root discriminant (GRD)''' of the field. Some results for low GRD fields can be found here.		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. NumberFields@home has helped establish the first known examples of imprimitive decic number fields with certain Galois groups. One can also consider ''first examples'' from the perspective of the '''Galois root discriminant (GRD)''' of the field. Some results for low GRD fields can be found here.

	==== Theoretical Physics ====		==== Theoretical Physics ====
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	== External links ==		== External links ==

	* NumberFields@~~Home~~ official website: numberfields.asu.edu/NumberFields/		* NumberFields@home official website: numberfields.asu.edu/NumberFields/
	* ASU Number Fields Database		* ASU Number Fields Database
	* LMFDB Number Fields search		* LMFDB Number Fields search

Al Piskun at 02:17, 30 May 2026

2026-05-30T02:17:19Z

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Al Piskun at 02:12, 30 May 2026

2026-05-30T02:12:48Z

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	{{~~cite web \|title=Test \|publisher=Test \|access-date=2026-05-29~~}}		{{Reflist}}

Al Piskun at 02:12, 30 May 2026

2026-05-30T02:12:26Z

← Older revision		Revision as of 02:12, 30 May 2026
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	~~[[wikipedia:Volunteer_computing~~\|~~volunteer computing]]~~		{{cite web \|title=Test \|publisher=Test \|access-date=2026-05-29}}

NumberFields@Home - Revision history

Al Piskun at 11:54, 31 May 2026

Al Piskun: re-add references

Al Piskun at 11:46, 30 May 2026

Admin: update and infobox

Al Piskun at 11:24, 30 May 2026

Admin at 11:21, 30 May 2026

Al Piskun at 03:01, 30 May 2026

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Al Piskun at 02:12, 30 May 2026