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- In [[mathematics]], and more specifically—in [[number theory]], an '''algebraic number''' is a [[complex number]] that is a root of a [[polynomial]] with [[ration ...s that ensued forms the foundation of modern [[algebraic number theory]]. Algebraic number theory is now an immense field, and one of current research, but so far has7 KB (1,145 words) - 00:49, 20 October 2013
- ...ging the statement "algebraic numbers can be complex" to the statement "an algebraic number is a complex number...", with the rationale that the second was correct and ...rs, but again, there is no canonical way to view this embedding. Thus, an algebraic number can generally be thought of as a complex number, but not in a canonical way7 KB (1,148 words) - 23:13, 10 December 2008
- 12 bytes (1 word) - 17:17, 10 January 2009
- ...x number]]s, but several recent authors have dropped this requirement. An algebraic number must be a root of a [[polynomial]] with [[rational number|rational]] coeffi ...s that ensued forms the foundation of modern [[algebraic number theory]]. Algebraic number theory is now an immense field, and one of current research, but so far has1 KB (179 words) - 14:14, 10 December 2008
- ...thor=A. Fröhlich | authorlink=Ali Fröhlich | coauthors=M.J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | pub *{{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebraic number theory | publisher=[[Springer-Verlag]] | isbn=0-387-94225-4 | year=1986 }}2 KB (209 words) - 02:28, 22 December 2008
- | pagename = Algebraic number | abc = Algebraic number774 bytes (74 words) - 16:30, 13 July 2008
- 12 bytes (1 word) - 10:54, 24 September 2007
- ...ory]], an '''algebraic number field''' is a principal object of study in [[algebraic number theory]]. The algebraic and arithmetic structure of a number field has app ...field]] '''Q''' of [[rational number]]s. The elements of ''K'' are thus [[algebraic number]]s. Let ''n'' = [''K'':'''Q'''] be the degree of the extension.7 KB (1,077 words) - 17:18, 10 January 2009
- 111 bytes (16 words) - 16:34, 13 July 2008
- ...n of the rational numbers of finite degree; a principal object of study in algebraic number theory.151 bytes (22 words) - 03:01, 1 January 2009
- {{r|Algebraic number theory}} {{r|Algebraic number field}}887 bytes (126 words) - 02:29, 22 December 2008
- ...'''cycle''') is a formal product of [[Place (mathematics)|place]]s of an [[algebraic number field]]. It is used to encode [[ramification]] data for [[abelian extensio Let ''K'' be an algebraic number field with ring of integers ''R''. A ''modulus'' is a formal product4 KB (561 words) - 20:25, 5 December 2008
- | pagename = algebraic number field | abc = algebraic number field2 KB (228 words) - 07:20, 15 March 2024
- 92 bytes (9 words) - 17:31, 27 October 2008
- | pagename = Modulus (algebraic number theory) | abc = Modulus (algebraic number theory)2 KB (233 words) - 17:30, 27 October 2008
- A formal product of places of an algebraic number field, used to encode ramification data for abelian extensions of a number167 bytes (25 words) - 15:54, 5 December 2008
- Auto-populated based on [[Special:WhatLinksHere/Algebraic number field]]. Needs checking by a human. {{r|Algebraic number}}843 bytes (113 words) - 10:49, 11 January 2010
- 12 bytes (1 word) - 01:20, 18 February 2009
- Auto-populated based on [[Special:WhatLinksHere/Modulus (algebraic number theory)]]. Needs checking by a human. {{r|Algebraic number field}}526 bytes (68 words) - 18:36, 11 January 2010
- ...field''' is an invariant attached to an [[field extension|extension]] of [[algebraic number field]]s which describes the geometric structure of the [[ring of integers]1 KB (235 words) - 01:20, 18 February 2009
- | pagename = Discriminant of an algebraic number field | abc = Discriminant of an algebraic number field864 bytes (73 words) - 07:19, 15 March 2024
- ...thor=A. Fröhlich | authorlink=Ali Fröhlich | coauthors=M.J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | pub * {{cite book | author=Gerald Janusz | title=Algebraic Number Fields | publisher=Academic Press | year=1973 | isbn=0-12-380520-4 }}1 KB (153 words) - 14:18, 16 January 2013
- An invariant attached to an extension of algebraic number fields which describes the geometric structure of the ring of integers and195 bytes (27 words) - 13:06, 23 December 2008
- Auto-populated based on [[Special:WhatLinksHere/Discriminant of an algebraic number field]]. Needs checking by a human. {{r|Algebraic number field}}554 bytes (72 words) - 16:00, 11 January 2010
Page text matches
- ...x number]]s, but several recent authors have dropped this requirement. An algebraic number must be a root of a [[polynomial]] with [[rational number|rational]] coeffi ...s that ensued forms the foundation of modern [[algebraic number theory]]. Algebraic number theory is now an immense field, and one of current research, but so far has1 KB (179 words) - 14:14, 10 December 2008
- ...theory]], '''class field theory''' studies the abelian extensions of an [[algebraic number field]], or more generally a [[global field]] or [[local field]].191 bytes (26 words) - 17:20, 10 January 2013
- Auto-populated based on [[Special:WhatLinksHere/Algebraic number field]]. Needs checking by a human. {{r|Algebraic number}}843 bytes (113 words) - 10:49, 11 January 2010
- {{r|Algebraic number theory}} {{r|Algebraic number field}}297 bytes (38 words) - 11:43, 15 June 2009
- * [[Discriminant of an algebraic number field]]352 bytes (43 words) - 04:36, 22 November 2023
- ...field extension|extension]] of [[algebraic number field]]s is a [[modulus (algebraic number theory)|modulus]] which determines the splitting of [[prime ideal]]s. If n For a general extension ''F''/''K'', the conductor is a [[modulus (algebraic number theory)|modulus]] of ''K''.1 KB (177 words) - 01:07, 18 February 2009
- Any [[subring]] of an [[algebraic number field]] composed of [[algebraic integer]]s forms an order: the ring of all307 bytes (47 words) - 13:58, 1 February 2009
- ...thor=A. Fröhlich | authorlink=Ali Fröhlich | coauthors=M.J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | pub * {{cite book | author=Gerald Janusz | title=Algebraic Number Fields | publisher=Academic Press | year=1973 | isbn=0-12-380520-4 }}1 KB (153 words) - 14:18, 16 January 2013
- * {{Citation | last=Weiss | first=Edwin | title=Algebraic number theory | publisher=Chelsea Publishing | year=1976 | isbn=0-8284-0293-0}}. ...ich | last2=Taylor | first2=Martin | authorlink2= Martin J. Taylor | title=Algebraic number theory | publisher=[[Cambridge University Press]] | series=Cambridge Studie470 bytes (55 words) - 09:40, 12 June 2009
- ...s]], a '''transcendental number''' is any [[complex number]] that is not [[algebraic number|algebraic]], i.e. it is not a root of any [[polynomial]] whose coefficients875 bytes (130 words) - 12:27, 8 May 2008
- #REDIRECT [[Algebraic number field#Unit group]]47 bytes (6 words) - 05:12, 1 January 2009
- #REDIRECT [[Algebraic number field#Unit group]]47 bytes (6 words) - 05:06, 1 January 2009
- {{r|Algebraic number theory}} {{r|Algebraic number field}}887 bytes (126 words) - 02:29, 22 December 2008
- #REDIRECT [[Discriminant of an algebraic number field]]55 bytes (7 words) - 13:09, 23 December 2008
- {{r|Algebraic number theory}}715 bytes (91 words) - 17:34, 10 December 2008
- *[[Algebraic number]]389 bytes (39 words) - 12:37, 4 January 2009
- #REDIRECT [[Modulus (algebraic number theory)#Ray class group]]63 bytes (8 words) - 06:18, 6 December 2008
- Generalization of the Riemann zeta function to algebraic number fields.107 bytes (13 words) - 07:50, 22 September 2008
- A computer algebra system for mathematicians interested in algebraic number theory.119 bytes (14 words) - 15:20, 28 October 2008
- Used in algebraic number theory; a modulus which determines the splitting of prime ideals.126 bytes (17 words) - 01:06, 18 February 2009
- An algebraic number field generated over the rational numbers by roots of unity.116 bytes (16 words) - 13:28, 7 December 2008
- A formal product of places of an algebraic number field, used to encode ramification data for abelian extensions of a number167 bytes (25 words) - 15:54, 5 December 2008
- | pagename = Algebraic number | abc = Algebraic number774 bytes (74 words) - 16:30, 13 July 2008
- Auto-populated based on [[Special:WhatLinksHere/Modulus (algebraic number theory)]]. Needs checking by a human. {{r|Algebraic number field}}526 bytes (68 words) - 18:36, 11 January 2010
- | pagename = Discriminant of an algebraic number field | abc = Discriminant of an algebraic number field864 bytes (73 words) - 07:19, 15 March 2024
- {{r|Algebraic number field}} {{r|Algebraic number}}1 KB (169 words) - 19:54, 11 January 2010
- {{r|Algebraic number}}276 bytes (34 words) - 10:41, 21 April 2010
- {{r|Algebraic number field}} {{r|Modulus (algebraic number theory)}}595 bytes (77 words) - 15:38, 11 January 2010
- {{r|Algebraic number field}} {{r|Algebraic number}}1 KB (146 words) - 16:32, 11 January 2010
- <noinclude>{{Subpages}}</noinclude>The branch of algebraic number theory which studies the abelian extensions of a number field, or more gene171 bytes (26 words) - 17:18, 10 January 2013
- An algebraic number field for which the ring of integers is a polynomial ring.114 bytes (17 words) - 17:08, 28 October 2008
- An invariant attached to an extension of algebraic number fields which encodes ramification data.133 bytes (17 words) - 17:23, 20 November 2008
- ...ted in algebraic number theory, performing sophisticated computations in [[algebraic number field]]s, in [[Global field|global]] [[function field]]s, and in [[local fi ...k | author=J. Graf von Schmettow | title=KANT — a tool for computations in algebraic number fields | booktitle=Computational number theory | publisher=de Gruyter | yea1 KB (152 words) - 08:31, 14 September 2013
- An element of an algebraic number field which has a denominator confined to primes in some fixed set.137 bytes (21 words) - 13:15, 5 December 2008
- {{r|Algebraic number field}} {{r|Discriminant of an algebraic number field}}857 bytes (112 words) - 16:32, 11 January 2010
- {{r|Algebraic number}}2 KB (206 words) - 19:38, 11 January 2010
- ...thor=A. Fröhlich | authorlink=Ali Fröhlich | coauthors=M.J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | pub *{{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebraic number theory | publisher=[[Springer-Verlag]] | isbn=0-387-94225-4 | year=1986 }}2 KB (209 words) - 02:28, 22 December 2008
- | title = Algebraic Number Theory240 bytes (22 words) - 07:44, 21 September 2008
- {{r|Algebraic number field}} {{r|Algebraic number}}592 bytes (76 words) - 20:06, 11 January 2010
- Auto-populated based on [[Special:WhatLinksHere/Discriminant of an algebraic number field]]. Needs checking by a human. {{r|Algebraic number field}}554 bytes (72 words) - 16:00, 11 January 2010
- In [[algebraic number theory]], the '''genus field''' ''G'' of a [[number field]] ''K'' is the [[ * {{cite book | last=Ishida | first=Makoto | title=The genus fields of algebraic number fields | series=Lecture Notes in Mathematics | publisher=[[Springer Verlag]846 bytes (124 words) - 16:14, 28 October 2008
- ...lgebraic, but the converse need not hold. For example, the field of all [[algebraic number]]s over '''Q''' is an algebraic extension but not of finite degree.3 KB (435 words) - 22:38, 22 February 2009
- * {{cite book | author=Pierre Samuel | authorlink=Pierre Samuel | title=Algebraic number theory | publisher=Hermann/Kershaw | year=1972 }}151 bytes (17 words) - 02:37, 4 January 2013
- {{r|Modulus (algebraic number theory)}}205 bytes (29 words) - 15:13, 10 January 2024
- ...n of the rational numbers of finite degree; a principal object of study in algebraic number theory.151 bytes (22 words) - 03:01, 1 January 2009
- ...field''' is an invariant attached to an [[field extension|extension]] of [[algebraic number field]]s which describes the geometric structure of the [[ring of integers]1 KB (235 words) - 01:20, 18 February 2009
- Roots of unity are clearly [[algebraic number]]s, and indeed [[algebraic integer]]s. It is often convenient to identify1 KB (197 words) - 22:01, 7 February 2009
- An invariant attached to an extension of algebraic number fields which describes the geometric structure of the ring of integers and195 bytes (27 words) - 13:06, 23 December 2008
- {{r|Algebraic number field}} {{r|Algebraic number}}2 KB (247 words) - 17:28, 11 January 2010
- In [[mathematics]], a '''monogenic field''' is an [[algebraic number field]] for which there exists an element In a monogenic field ''K'', the [[Discriminant of an algebraic number field|field discriminant]] of ''K'' is equal to the [[discriminant of a pol1 KB (208 words) - 16:47, 17 December 2008
- ...c embedding of the generators of the unit group of the maximal order of an algebraic number field.168 bytes (25 words) - 05:11, 1 January 2009
- ...ink=J. W. S. Cassels | coauthors=[[Albrecht Fröhlich|A. Fröhlich]] | title=Algebraic Number Theory | publisher=[[Academic Press]] | year=1967 | isbn=012268950X }} * {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebraic number theory | publisher=[[Springer-Verlag]] | isbn=0-387-94225-4 | year=1986 }}865 bytes (110 words) - 02:29, 10 January 2013
- ...ink=J. W. S. Cassels | coauthors=[[Albrecht Fröhlich|A. Fröhlich]] | title=Algebraic Number Theory | publisher=[[Academic Press]] | year=1967 | isbn=012268950X }} * {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebraic number theory | publisher=[[Springer-Verlag]] | isbn=0-387-94225-4 | year=1986 }}865 bytes (110 words) - 17:22, 10 January 2013
- {{r|Algebraic number}}564 bytes (72 words) - 16:08, 11 January 2010
- {{r|Algebraic number field}} {{r|Algebraic number}}1 KB (187 words) - 20:18, 11 January 2010
- In [[mathematics]], and more specifically—in [[number theory]], an '''algebraic number''' is a [[complex number]] that is a root of a [[polynomial]] with [[ration ...s that ensued forms the foundation of modern [[algebraic number theory]]. Algebraic number theory is now an immense field, and one of current research, but so far has7 KB (1,145 words) - 00:49, 20 October 2013
- {{r|Algebraic number}}2 KB (247 words) - 06:00, 7 November 2010
- {{r|Algebraic number field}} {{r|Algebraic number}}2 KB (262 words) - 19:07, 11 January 2010
- {{r|Algebraic number}}454 bytes (55 words) - 03:14, 21 October 2010
- {{r|Discriminant of an algebraic number field}}136 bytes (19 words) - 11:05, 31 May 2009
- {{r|Algebraic number}}566 bytes (73 words) - 16:56, 11 January 2010
- ...e of integral closure is the [[ring of integers]] or maximal order in an [[algebraic number field]] ''K'', which may be defined as the integral closure of '''Z''' in ' * {{cite book | author=Pierre Samuel | authorlink=Pierre Samuel | title=Algebraic number theory | publisher=Hermann/Kershaw | year=1972 }}1 KB (172 words) - 15:42, 7 February 2009
- ...], the '''different ideal''' is an invariant attached to an extension of [[algebraic number field]]s. ...tive norm]] of the relative different is equal to the [[Discriminant of an algebraic number field|relative discriminant]] Δ<sub>''L''/''K''</sub>. In a tower of fiel2 KB (382 words) - 09:40, 12 June 2009
- ...of a [[square matrix]], an [[endomorphism]] of a [[vector space]] or an [[algebraic number]]. ==Minimal polynomial of an algebraic number==4 KB (613 words) - 02:34, 4 January 2013
- ...'''cycle''') is a formal product of [[Place (mathematics)|place]]s of an [[algebraic number field]]. It is used to encode [[ramification]] data for [[abelian extensio Let ''K'' be an algebraic number field with ring of integers ''R''. A ''modulus'' is a formal product4 KB (561 words) - 20:25, 5 December 2008
- {{r|Algebraic number field}}584 bytes (79 words) - 15:48, 11 January 2010
- {{r|Algebraic number field}}675 bytes (89 words) - 17:28, 11 January 2010
- {{r|Algebraic number field}}644 bytes (86 words) - 19:50, 11 January 2010
- | pagename = algebraic number field | abc = algebraic number field2 KB (228 words) - 07:20, 15 March 2024
- ...ging the statement "algebraic numbers can be complex" to the statement "an algebraic number is a complex number...", with the rationale that the second was correct and ...rs, but again, there is no canonical way to view this embedding. Thus, an algebraic number can generally be thought of as a complex number, but not in a canonical way7 KB (1,148 words) - 23:13, 10 December 2008
- | pagename = Modulus (algebraic number theory) | abc = Modulus (algebraic number theory)2 KB (233 words) - 17:30, 27 October 2008
- {{rpl|Algebraic number field}} {{rpl|Discriminant of an algebraic number field}}5 KB (628 words) - 04:35, 22 November 2023
- {{r|Algebraic number field}}432 bytes (56 words) - 17:48, 11 January 2010
- In [[mathematics]], to each [[algebraic number field]] ''k'', there is associated an important function called the '''Dede If ''k'' is an algebraic number field, the Dedekind zeta function of the field is a [[meromorphic function]2 KB (343 words) - 07:23, 1 January 2009
- ...ory]], an '''algebraic number field''' is a principal object of study in [[algebraic number theory]]. The algebraic and arithmetic structure of a number field has app ...field]] '''Q''' of [[rational number]]s. The elements of ''K'' are thus [[algebraic number]]s. Let ''n'' = [''K'':'''Q'''] be the degree of the extension.7 KB (1,077 words) - 17:18, 10 January 2009
- {{r|Algebraic number field}}472 bytes (61 words) - 11:04, 11 January 2010
- {{r|Algebraic number field}}497 bytes (63 words) - 17:28, 11 January 2010
- {{r|Algebraic number field}}476 bytes (61 words) - 18:38, 11 January 2010
- ...thor=A. Fröhlich | authorlink=Ali Fröhlich | coauthors=M.J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | pub * {{cite book | author=Pierre Samuel | authorlink=Pierre Samuel | title=Algebraic number theory | publisher=Hermann/Kershaw | year=1972 }}2 KB (342 words) - 12:52, 21 January 2009
- {{r|Algebraic number field}}502 bytes (64 words) - 19:15, 11 January 2010
- {{r|Algebraic number}}544 bytes (70 words) - 18:34, 11 January 2010
- ...thor=A. Fröhlich | authorlink=Ali Fröhlich | coauthors=M.J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | pub ...art | authorlink=Ian Stewart (mathematician) | coauthors=D.O. Tall | title=Algebraic number theory | publisher=Chapman and Hall | year=1979 | isbn=0-412-13840-9 | page3 KB (453 words) - 17:18, 6 February 2009
- {{r|Algebraic number field}}1 KB (174 words) - 20:03, 11 January 2010
- In [[mathematics]], in the field of [[algebraic number theory]], an '''''S'''''<nowiki></nowiki>'''-unit''' generalises the idea o *{{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebraic number theory | publisher=Springer | isbn=0-387-94225-4 | year=1986 }} Chap. V.3 KB (381 words) - 16:02, 28 October 2008
- {{r|Algebraic number field}}762 bytes (99 words) - 17:00, 11 January 2010
- ...athbb{Q}</math>, i.e. the field of roots of rational polynomials, is the [[algebraic number]]s.9 KB (1,446 words) - 08:52, 30 May 2009
- {{r|Algebraic number}}850 bytes (136 words) - 15:22, 31 October 2008
- {{r|Algebraic number theory}}857 bytes (137 words) - 16:47, 30 October 2008
- {{r|Algebraic number theory}}924 bytes (147 words) - 17:24, 10 January 2013
- The term '''rational integer''' is used in [[algebraic number theory]] to distinguish these "ordinary" integers, embedded in the [[field10 KB (1,566 words) - 08:34, 2 March 2024
- {{r|Algebraic number field}}1 KB (169 words) - 08:53, 22 December 2008
- ...These include the [[unique factorization|unique factorization theorem]], [[algebraic number fields]], [[elliptic curves]], and [[modular form]]s.2 KB (340 words) - 12:36, 22 February 2012
- ...pace, there is an associated Artin L-function. When ''K'' and ''k'' are [[algebraic number field]]s, Artin L-functions generalize [[Dedekind zeta function]]s, which a2 KB (315 words) - 15:49, 10 December 2008
- ...algebraic structures, such as algebraic topology, algebraic geometry, and algebraic number theory. A strong understanding of module theory is essential for anyone de7 KB (1,154 words) - 02:39, 16 May 2009
- #Let <math>K</math> be an [[algebraic number field]]. Then the integral closure <math>o_K</math>of <math>\mathbb{Z}</mat2 KB (306 words) - 15:51, 10 December 2008
- ...the study of [[polynomial|polynomial rings]] and [[Algebraic number field|algebraic number fields]] in the second half of the nineteenth century, amongst other by [[R10 KB (1,667 words) - 13:47, 5 June 2011
- Let ''K'' be an [[algebraic number field]], a finite [[field extension|extension]] of '''Q''', and ''E'' an el10 KB (1,637 words) - 16:03, 17 December 2008
- :::I have to admit I planned on adding some articles to algebraic number theory before treating things like elliptic curves and modular forms, so an ...ave been written might be difficult though. Anyhow, I want to have a go at algebraic number theory first. [[User:William Hart|William Hart]] 19:59, 23 February 2007 (C13 KB (2,289 words) - 08:03, 6 July 2007
- ...[[rational number|rational]] or [[irrational number|irrational]]; either [[algebraic number|algebraic]] or [[transcendental number|transcendental]]; and either [[posit ...real numbers is [[uncountable|uncountably infinite]] but the set of all [[algebraic number]]s is [[countable|countably infinite]]. Contrary to widely held beliefs, hi19 KB (2,948 words) - 10:07, 28 February 2024
- ...odern subfields - in particular, [[analytic number theory|analytic]] and [[algebraic number theory]]. Algebraic number theory may be said to start with the study of reciprocity and cyclotomy, bu27 KB (4,383 words) - 08:05, 11 October 2011