Jump to content
Main menu
Main menu
move to sidebar
hide
Navigation
Main page
Recent changes
Random page
Help about MediaWiki
Special pages
Niidae Wiki
Search
Search
Appearance
Create account
Log in
Personal tools
Create account
Log in
Pages for logged out editors
learn more
Contributions
Talk
Editing
Field (mathematics)
(section)
Page
Discussion
English
Read
Edit
View history
Tools
Tools
move to sidebar
hide
Actions
Read
Edit
View history
General
What links here
Related changes
Page information
Appearance
move to sidebar
hide
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
== References == {{refbegin|30em}} * {{citation |title=Introduction to Field Theory| last=Adamson|first=I. T.|isbn=978-0-486-46266-0|year=2007|publisher=Dover Publications}} * {{citation | last1=Allenby | first1=R. B. J. T. | title=Rings, Fields and Groups | publisher=Butterworth-Heinemann | isbn=978-0-340-54440-2 | year=1991}} * {{citation | last1=Artin | first1=Michael | author1-link=Michael Artin | title=Algebra | publisher=[[Prentice Hall]] | isbn=978-0-13-004763-2 | year=1991}}, especially Chapter 13 * {{citation |last1=Artin|first1=Emil|last2=Schreier|first2=Otto|author1-link=Emil Artin|author2-link=Otto Schreier|title=Eine Kennzeichnung der reell abgeschlossenen Körper|journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg|issn=0025-5858| volume=5|pages=225–231|year=1927|language=de|doi=10.1007/BF02952522|jfm=53.0144.01|s2cid=121547404}} * {{citation | last=Ax |first=James|author-link=James Ax|year=1968|title=The elementary theory of finite fields|journal=Ann. of Math. |series= 2|volume=88|issue=2|pages=239–271|doi=10.2307/1970573|jstor=1970573}} * {{citation |last=Baez|first=John C.|author-link=John C. Baez|title=The octonions|journal=[[Bulletin of the American Mathematical Society]] |volume=39|issue=2|year=2002|pages=145–205|doi=10.1090/S0273-0979-01-00934-X|arxiv=math/0105155|s2cid=586512}} * {{citation | last=Banaschewski|first=Bernhard|title=Algebraic closure without choice.|year=1992|journal=Z. Math. Logik Grundlagen Math.|volume=38|issue=4|pages=383–385|zbl=0739.03027|doi=10.1002/malq.19920380136}} * {{citation |last1=Beachy|first1=John. A|last2=Blair|first2=William D.|title=Abstract Algebra|edition=3|publisher=Waveland Press|isbn=1-57766-443-4|year=2006}} * {{citation | last1=Blyth | first1=T. S. | last2=Robertson | first2=E. F. |author2-link=Edmund F. Robertson| title=Groups, rings and fields: Algebra through practice| publisher=[[Cambridge University Press]] | year=1985}}. See especially Book 3 ({{ISBN|0-521-27288-2}}) and Book 6 ({{ISBN|0-521-27291-2}}). * {{citation |last1=Borceux|first1=Francis|last2=Janelidze|first2=George|title=Galois theories| isbn=0-521-80309-8|year=2001|publisher=Cambridge University Press|zbl=0978.12004}} * {{citation |last1=Bourbaki|first1=Nicolas|author-link=Nicolas Bourbaki|title=Elements of the history of mathematics|publisher=Springer|year=1994|isbn=3-540-19376-6|mr=1290116|doi=10.1007/978-3-642-61693-8}} * {{citation |title=Algebra II. Chapters 4–7|last=Bourbaki|first=Nicolas|author-link=Nicolas Bourbaki|isbn=0-387-19375-8|year=1988|publisher=Springer}} * {{citation |last=Cassels|first=J. W. S.|author-link=J. W. S. Cassels|title=Local fields|series=London Mathematical Society Student Texts|volume=3|publisher=Cambridge University Press|year=1986|isbn=0-521-30484-9|mr=861410|doi=10.1017/CBO9781139171885}} * {{citation |title=Elements of Abstract Algebra|last=Clark|first=A.|isbn=978-0-486-64725-8|series=Dover Books on Mathematics Series|year=1984|publisher=Dover |url=https://books.google.com/books?id=bj1kOY8gOfcC }} * {{citation |last1=Conway |first1=John Horton |year=1976 |author-link1=John Horton Conway |title=On Numbers and Games |publisher=[[Academic Press]] |title-link=On Numbers and Games }} * {{citation |last=Corry|first=Leo|author-link=Leo Corry|title=Modern algebra and the rise of mathematical structures|edition=2nd|isbn=3-7643-7002-5|year=2004|publisher=Birkhäuser|zbl=1044.01008}} * {{citation |last1=Dirichlet|first1=Peter Gustav Lejeune|author-link=Peter Gustav Lejeune Dirichlet|editor-last=Dedekind|editor-first=Richard|editor-link=Richard Dedekind|year=1871|edition=2nd|language=de|volume=1|location=Braunschweig, Germany|publisher=Friedrich Vieweg und Sohn|title=Vorlesungen über Zahlentheorie (Lectures on Number Theory)|url=https://books.google.com/books?id=SRJTAAAAcAAJ&pg=PA424}} * {{citation | last=Eisenbud|first=David|author-link=David Eisenbud | title=Commutative algebra with a view toward algebraic geometry | location=New York | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics]] | volume=150 | year=1995 | mr=1322960 | isbn=0-387-94268-8 | doi=10.1007/978-1-4612-5350-1}} * {{citation |last=Escofier|first=J. P.|isbn=978-1-4613-0191-2|title=Galois Theory|publisher=Springer|year=2012}} * {{citation | last1 = Fraleigh | first1 = John B. | year = 1976 | isbn = 0-201-01984-1 | title = A First Course In Abstract Algebra | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading }} * {{citation | last1= Fricke | first1= Robert|author1-link=Robert Fricke | last2= Weber | first2= Heinrich Martin | author2-link= Heinrich Martin Weber | title= Lehrbuch der Algebra | language=de|url= http://resolver.sub.uni-goettingen.de/purl?PPN234788267 | publisher= Vieweg | year= 1924 | jfm= 50.0042.03}} * {{citation |title=''p''-adic numbers| last=Gouvêa|first=Fernando Q.|author-link=Fernando Q. Gouvêa|edition=2nd|year=1997|publisher=Springer|series=Universitext}} * {{citation |title=A Guide to Groups, Rings, and Fields| last=Gouvêa|first=Fernando Q.|author-link=Fernando Q. Gouvêa|isbn=978-0-88385-355-9|year=2012|publisher=Mathematical Association of America}} * {{springer|title=Field|id=p/f040090}} * {{citation |last1=Hensel|first1=Kurt|author1-link=Kurt Hensel|title=Über eine neue Begründung der Theorie der algebraischen Zahlen|journal=Journal für die Reine und Angewandte Mathematik| issn=0075-4102|volume=128|pages=1–32|year=1904|language=de|jfm=35.0227.01|url=https://eudml.org/doc/149187}} * {{citation | last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 1 | publisher=Dover| isbn = 978-0-486-47189-1}} * {{citation |mr=0679774|first1=Uwe|last1=Jannsen|first2=Kay|last2=Wingberg|title=Die Struktur der absoluten Galoisgruppe 𝔭-adischer Zahlkörper. [The structure of the absolute Galois group of 𝔭-adic number fields]|journal=Invent. Math.|volume=70|year=1982|issue=1|pages=71–98|url=http://epub.uni-regensburg.de/26689/|doi=10.1007/bf01393199|bibcode=1982InMat..70...71J|s2cid=119378923}} * {{citation |last=Kleiner|first=Israel|editor1-first=Israel|editor1-last=Kleiner|author-link=Israel Kleiner (mathematician)|title=A history of abstract algebra|publisher=Birkhäuser|year=2007|isbn=978-0-8176-4684-4|mr=2347309|doi=10.1007/978-0-8176-4685-1}} * {{citation |last=Kiernan|first=B. Melvin|title=The development of Galois theory from Lagrange to Artin|journal=Archive for History of Exact Sciences |volume=8|year=1971|number=1–2|pages=40–154|mr=1554154|doi=10.1007/BF00327219|s2cid=121442989}} * {{citation |last1=Kuhlmann|first1=Salma|title=Ordered exponential fields|series=Fields Institute Monographs|volume=12|publisher=American Mathematical Society|year=2000|isbn=0-8218-0943-1|mr=1760173}} * {{citation |first=Serge|last=Lang|author-link=Serge Lang|title=Algebra|series=Graduate Texts in Mathematics|volume=211|edition=3rd|publisher=Springer|year=2002|isbn=0-387-95385-X|doi=10.1007/978-1-4613-0041-0}} * {{citation |first1=Rudolf|last1=Lidl|first2=Harald|last2=Niederreiter|author2-link=Harald Niederreiter|title=Finite fields| edition=2nd|year=2008|isbn=978-0-521-06567-2|publisher=Cambridge University Press|zbl=1139.11053}} * {{citation |last=Lorenz|first=Falko|title=Algebra, Volume II: Fields with Structures, Algebras and Advanced Topics|year=2008|isbn=978-0-387-72487-4|publisher=Springer}} * {{citation |last1=Marker|first1=David|last2=Messmer|first2=Margit|last3=Pillay|first3=Anand|title=Model theory of fields|series=Lecture Notes in Logic|volume=5|edition=2nd|publisher=Association for Symbolic Logic|year=2006|isbn=978-1-56881-282-3|mr=2215060|citeseerx=10.1.1.36.8448|url-access=registration|url=https://archive.org/details/modeltheoryoffie0000mark}} * {{citation | last1 = McCoy | first1 = Neal H. | title = Introduction To Modern Algebra, Revised Edition | location = Boston | publisher = [[Allyn and Bacon]] | year = 1968 | lccn = 68015225 }} * {{citation |last1=Mines|first1=Ray|last2=Richman|first2=Fred|last3=Ruitenburg|first3=Wim|title=A course in constructive algebra|series=Universitext|publisher=Springer|year=1988|isbn=0-387-96640-4|mr=919949|doi=10.1007/978-1-4419-8640-5}} * {{citation | last = Moore | first = E. Hastings | author-link = E. H. Moore | year = 1893 | title = A doubly-infinite system of simple groups | journal = [[Bulletin of the American Mathematical Society]] | volume = 3 | issue = 3 | pages = 73–78 | mr = 1557275 | doi = 10.1090/S0002-9904-1893-00178-X | doi-access = free }} * {{citation |last=Prestel|first=Alexander|title=Lectures on Formally Real Fields |series=Lecture Notes in Mathematics |volume=1093|publisher=Springer|year=1984|isbn=3-540-13885-4|mr=769847|doi=10.1007/BFb0101548}} * {{citation |last=Ribenboim|first=Paulo|author-link=Paulo Ribenboim|title=The theory of classical valuations|series=Springer Monographs in Mathematics|publisher=Springer|year=1999|isbn=0-387-98525-5|mr=1677964| doi=10.1007/978-1-4612-0551-7}} * {{citation |last=Scholze|first=Peter|author-link=Peter Scholze|chapter=Perfectoid spaces and their Applications|year=2014|chapter-url=http://www.math.uni-bonn.de/people/scholze/ICM.pdf|title=Proceedings of the International Congress of Mathematicians 2014|publisher=Kyung Moon SA |url=http://www.icm2014.org/en/vod/proceedings.html|isbn=978-89-6105-804-9}} * {{citation |last=Schoutens|first=Hans|isbn=978-3-642-13367-1|title=The Use of Ultraproducts in Commutative Algebra|year=2002|publisher=Springer|series=Lecture Notes in Mathematics|volume=1999}} * {{citation |last=Serre|first=Jean-Pierre|author-link=Jean-Pierre Serre|title=A course in arithmetic. Translation of ''Cours d'arithmetique''|edition=2nd|orig-year=1978|series=Graduate Text in Mathematics|volume=7|publisher=Springer|zbl=0432.10001|isbn=9780387900407|year=1996|url-access=registration|url=https://archive.org/details/courseinarithmet00serr}} * {{citation|last=Serre|first=Jean-Pierre|author-link=Jean-Pierre Serre|title=Local fields|series=Graduate Texts in Mathematics |volume=67|publisher=Springer|year=1979|isbn=0-387-90424-7|mr=554237}} * {{citation|first=Jean-Pierre|last=Serre|author-link=Jean-Pierre Serre|title=Topics in Galois theory|isbn=0-86720-210-6|year=1992|publisher=Jones and Bartlett Publishers|zbl=0746.12001}} * {{citation | last1=Serre | first1=Jean-Pierre | author1-link= Jean-Pierre Serre | title=Galois cohomology | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-42192-4 | mr=1867431 | year=2002 | zbl=1004.12003 | others=Translated from the French by [[Patrick Ion]]}} * {{citation|last=Sharpe|first=David|title=Rings and factorization|isbn=0-521-33718-6|year=1987|publisher=Cambridge University Press|zbl=0674.13008|url-access=registration|url=https://archive.org/details/ringsfactorizati0000shar}} * {{citation | last1=Steinitz | first1=Ernst | author1-link=Ernst Steinitz | year=1910 | title=Algebraische Theorie der Körper |trans-title=Algebraic Theory of Fields | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002167042 | journal=[[Journal für die reine und angewandte Mathematik]] | volume=1910 | issue=137 | pages=167–309 | issn=0075-4102 | jfm=41.0445.03 | doi=10.1515/crll.1910.137.167 | s2cid=120807300 }} * {{citation | last1 = Tits | first1 = Jacques | author-link=Jacques Tits | chapter = Sur les analogues algébriques des groupes semi-simples complexes | title = Colloque d'algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques Établissements Ceuterick, Louvain | publisher = Librairie Gauthier-Villars | place = Paris | year = 1957 | pages = 261–289 }} * {{citation |title=Galois Theory of Linear Differential Equations|first1=M.|last1=van der Put|first2=M. F.|last2=Singer|year=2003|publisher=Springer|series=Grundlehren der mathematischen Wissenschaften|volume=328|url=http://www4.ncsu.edu/~singer/papers/dbook2.ps}} * {{citation | last=von Staudt|first=Karl Georg Christian|author-link=Karl Georg Christian von Staudt |url=https://books.google.com/books?id=XwEHAAAAcAAJ&pg=PA127|title=Beiträge zur Geometrie der Lage (Contributions to the Geometry of Position)|volume=2|location=Nürnberg (Germany)|publisher=Bauer and Raspe|year=1857}} * {{citation |last=Wallace|first=D. A. R.|year=1998|title=Groups, Rings, and Fields|series=SUMS|publisher=Springer|volume=151}} * {{citation |first=Seth|last=Warner|title=Topological fields|isbn=0-444-87429-1|year=1989|publisher=North-Holland|zbl=0683.12014}} * {{citation |first=Lawrence C.|last= Washington|author-link=Lawrence C. Washington|title=Introduction to Cyclotomic Fields|series=Graduate Texts in Mathematics|volume= 83|publisher=Springer-Verlag |year= 1997|edition=2nd |isbn=0-387-94762-0 |mr=1421575|doi=10.1007/978-1-4612-1934-7}} * {{citation |last=Weber |first=Heinrich |author-link= Heinrich Martin Weber |title=Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie |journal=Mathematische Annalen |issn=0025-5831 |volume=43 |issue=4 |pages=521–549 |year=1893 |language=de |doi=10.1007/BF01446451 |jfm=25.0137.01 |s2cid=120528969 |url=https://eudml.org/doc/157689 }} {{refend}}
Summary:
Please note that all contributions to Niidae Wiki may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here.
You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see
Encyclopedia:Copyrights
for details).
Do not submit copyrighted work without permission!
Cancel
Editing help
(opens in new window)
Search
Search
Editing
Field (mathematics)
(section)
Add topic