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class="display-6">About: <a href="http://dbpedia.org/resource/Field_(mathematics)">Field (mathematics)</a> </h1> </div> </div> <div class="row"> <div class="col"> <div class="text-muted"> An Entity of Type: <a href="http://dbpedia.org/ontology/Building">building</a>, from Named Graph: <a href="http://dbpedia.org">http://dbpedia.org</a>, within Data Space: <a href="http://dbpedia.org">dbpedia.org</a> </div> </div> </div> <div class="row pt-2"> <div class="col-xs-9 col-sm-10"> In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. </div> <div class="col-xs-3 col-sm-2"> <a href="#" class="thumbnail"> <img src="http://commons.wikimedia.org/wiki/Special:FilePath/Regular_polygon_7_annotated.svg?width=300" alt="thumbnail" class="img-fluid" /> </a> </div> </div> </div> </section>   <section> <div class="container-xl"> <div class="row"> <div class="table-responsive"> <table class="table table-hover table-sm table-light"> <thead> <tr> <th class="col-xs-3 ">Property</th> <th class="col-xs-9 px-3">Value</th> </tr> </thead> <tbody> <tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/abstract">dbo:abstract</a> </td><td class="col-10 text-break"><ul> <li style="display:none;">En l'àlgebra abstracta, un cos és un sistema algebraic en què és possible efectuar la suma, resta, multiplicació i divisió (llevat de la divisió per 0), i en la qual se satisfan certes lleis. (ca)</li> <li style="display:none;">Komutativní těleso (někdy stručně těleso podle německého körper, někdy též pole z anglického field) je algebraická struktura, na které jsou definovány dvě binární operace, sčítání a násobení, pro které platí řada určených vlastností. Jedná se o taková tělesa, kde násobení splňuje navíc komutativitu, respektive takové , kde navíc existuje inverzní prvek pro obě binární operace (okruh vyžadoval existenci inverzního prvku jen pro operaci +). Tělesa, ve kterých násobení není komutativní, se nazývají . (cs)</li> <li style="display:none;">في الرياضيات، الحقل (بالإنجليزية: Field)‏ هي مجموعة عُرفت عليها أربع عمليات هن الجمع والطرح والضرب والقسمة ويحقق خواص ما يقابلها من عمليات الأعداد الحقيقة والنسبية. فبالتالي الحقل هو بنية أساسية في الجبر ويستخدم بشكل واسع في الجبر ونظرية الأعداد وغيره من فروع الرياضيات. أفضل الحقول المعروفه هي حقل الأعداد النسبية وحقل الأعداد الحقيقة وكذلك حقل الأعداد المركبة. هناك أيضا العديد من الحقول الأخرى مثل حقول الدوال النسبية وحقول الدوال الجبرية (algebraic function field ) وحقول الأعداد الجبرية وكذلك حقول p-adic . هذه الحقول هي الأكثر استخداما في دراسة الرياضيات خصوصا في نظرية الأعداد والهندسة الجبرية. كثير من بروتوكلات التشفير تعتمد على الحقول المنتهية والتي نعني بها تحتوي على العديد من العناصرة المنتهية. يمكن التعبير عن العلاقة أو الصلة بين حقلين بواسطة مايسمى امتداد الحقول. طورت نظرية غالوا، والتي أكتشفها العالم إيفاريست غالوا في ثلاثينات القرن التاسع عشر، لفهم العلاقات المتقدمة. نظريات أساسية في التحليل مرتبطة بالخواص الهيكلية لحقل الأعداد الحققية.والأكثر أهمية لأسباب جبرية، أنه أي حقل ممكن أن يستخدم كمية قياسية في فضاء المتجه والذي يعتبر محتوى أساسي عام للجبر الخطي. تشترك حقول الأعداد مع حقل الأعداد النسبية والتي تُدرس بعمق في نظرية الأعداد. يمكن لحقول الدوال أن تساعد في وصف خواص الأشياء الهندسية. (ar)</li> <li style="display:none;">Σώμα (από το γαλλικό Corps) είναι ένα σύνολο (από το αγγλικό Field) αντικειμένων οποιουδήποτε είδους, μαζί με δύο δυαδικές πράξεις + και * ορισμένες στο , οι οποίες απεικονίζουν 2 στοιχεία a και b που ανήκουν στο F στα a+b και a*b, επίσης στοιχεία του F.Και ισχύουν οι εξής ιδιότητες: 1. * 2. * (υπάρχει στοιχείο 0 που ανήκει στο F), τέτοιο ώστε * για κάθε που ανήκει στο , και * (για κάθε a που ανήκει στο F υπάρχει b που ανήκει στο F τέτοιο ώστε a+b=0). 1. * Δηλαδή να ισχύει η αντιμεταθετική ιδιότητα στο F 2. * 3. * Υπάρχει αριθμός 1 που ανήκει στο F τέτοιος ώστε (i).a*1=a (ii). Και να υπάρχει, για κάθε a διάφορο του μηδενός, ένα b, τέτοιο ώστε a*b=1. 4. * 5. * Τα γνωστά παραδείγματα σωμάτων όπως είναι προφανές από τα θεωρήματα του Σώματος είναι το και το και το σώμα των μιγαδικών αριθμών .Βεβαίως τα + και το * είναι τα γνωστά σύμβολα της πρόσθεσης και του πολλαπλασιασμού άρα δεν χρειάζονται περαιτέρω διερεύνηση.Το στοιχείο 0 είναι το ουδέτερο στοιχείο της πρόσθεσης και το 1 είναι το ουδέτερο στοιχείο του πολλαπλασιασμού.Το αντίθετο της πρόσθεσης το συμβολίζουμε με -a έτσι ώστε για κάθε a να υπάρχει -a, τέτοιο ώστε a+(-a)=0, και το αντίστροφο του πολλαπλασιασμού συμβολίζεται με , τέτοιο ώστε, για κάθε a που ανήκει στο F, να υπάρχει τέτοιο ώστε a* =1. Εκτός από τα γνωστά παραδείγματα σωμάτων υπάρχουν και τα παραδείγματα των σωμάτων που είναι της μορφής a+b* και γενικά της μορφής αυτής που το υπόρριζο μπορεί να πάρει τις τιμές 2,3,...,ν. Ένας δακτύλιος καλείται σώμα αν ισχύουν τα εξής : * Ο δακτύλιος είναι μεταθετικός. * Υπάρχει Μοναδιαίο Στοιχείο ώστε για κάθε * Για κάθε υπάρχει στοιχείο του το οποίο συμβολίζουμε με τέτοιο ώστε Τυπικό παράδειγμα σώματος είναι το σύνολο των πραγματικών αριθμών , καθώς είναι μοναδιαίος αντιμεταθετικός δακτύλιος και κάθε μη μηδενικό στοιχείο του έχει αντίστροφο. (el)</li> <li style="display:none;">En matematiko kaj, pli specife, en abstrakta algebro kampo estas komuta korpo. Tio estas unu el la plej gravaj nocioj de multaj fakoj de abstrakta algebro kaj nombro-teorio. Pli detale, oni povas karakterizi la nocion kampo K per ĉi-subaj aksiomoj. (eo)</li> <li style="display:none;">En matemática, concretamente en el campo de la álgebra abstracta, un cuerpo (en ocasiones llamado campo como traducción de inglés field) es un sistema algebraico en el cual las operaciones llamadas adición y multiplicación se pueden realizar y cumplen las propiedades: asociativa, conmutativa y distributiva de la multiplicación respecto de la adición, además de la existencia de inverso aditivo, de inverso multiplicativo y de un elemento neutro para la adición y otro para la multiplicación, los cuales permiten efectuar las operaciones de sustracción y división (excepto la división por cero); estas propiedades ya son familiares de la aritmética de números racionales. Los cuerpos son estructuras algebraicas importantes de estudio en diversas ramas de las matemáticas puras: álgebra abstracta, análisis matemático, teoría de números, geometría, topología, física matemática, etc.; puesto que proporcionan generalizaciones apropiadas de operaciones binarias en conjuntos y sistemas de números tales como los conjuntos de números racionales, números reales y números complejos. El concepto de un cuerpo se usa, por ejemplo, al definir y construir formalmente un espacio vectorial y las transformaciones en estos objetos, dadas por matrices, objetos en el álgebra lineal cuyos componentes pueden ser elementos de un cuerpo arbitrario. La teoría de Galois estudia las relaciones de simetría en las ecuaciones algebraicas, desde la observación del comportamiento de sus raíces y las extensiones de cuerpos correspondientes y su relación con los automorfismos de cuerpos correspondientes. (es)</li> <li>In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle cannot be done with a compass and straightedge. Moreover, it shows that quintic equations are, in general, algebraically unsolvable. Fields serve as foundational notions in several mathematical domains. This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory. Function fields can help describe properties of geometric objects. (en)</li> <li style="display:none;">Ein Körper ist im mathematischen Teilgebiet der Algebra eine ausgezeichnete algebraische Struktur, in der die Addition, Subtraktion, Multiplikation und Division auf eine bestimmte Weise durchgeführt werden können. Die Bezeichnung „Körper“ wurde im 19. Jahrhundert von Richard Dedekind eingeführt. Die wichtigsten Körper, die in fast allen Gebieten der Mathematik benutzt werden, sind der Körper der rationalen Zahlen, der Körper der reellen Zahlen und der Körper der komplexen Zahlen. (de)</li> <li style="display:none;">Aljebra abstraktuan gorputza da multzoak gutxienez bi elementu izanda, multzorako (gehiketa) eragiketak elkartze eta trukatze propietatea eta elementu alderantzizko eta neutroaren existentzia betetzen dituen, eta (biderketa edo produktua)eragiketak elkartze eta banatze propietateak, elementu neutroaren existentzia eta multzoko elementuentzako, gehiketarekiko elementu neutroa izan ezik, elementu alderantzizkoaren existentzia betetzen dituen egitura aljebraikoa. (eu)</li> <li style="display:none;">Lapangan atau medan (juga disebut bidang) dalam matematika adalah suatu struktur aljabar dengan operasi seperti penambahan, pengurangan, perkalian, dan pembagian yang memenuhi aksioma tertentu. Lapangan yang kerap kali dijumpai adalah lapangan bilangan riil, lapangan bilangan kompleks dan bilangan rasional. Medan yang paling dikenal adalah medan bilangan rasional, bidang bilangan riil dan medan bilangan kompleks. Terdapat medan lainnya, seperti , , medan bilangan aljabar, dan umumnya digunakan dan dipelajari dalam matematika, terutama dalam teori bilangan dan geometri aljabar. Sebagian besar protokol kriptografi mengandalkan Medan hingga, yaitu bidang dengan banyak . Relasi dua medan diekspresikan dengan gagasan tentang . Teori Galois yang diprakarsai oleh Évariste Galois pada tahun 1830-an, dikhususkan untuk memahami kesimetrian perluasan medan. Di antara hasil lainnya, teori ini menunjukkan bahwa dan tidak dapat dilakukan dengan . Selain itu, ini menunjukkan bahwa persamaan kuintik, secara umum, tidak berpenyelesaian secara aljabar. Medan berfungsi sebagai gagasan dasar dalam beberapa ranah matematika. Ini mencakup berbagai cabang analisis matematika yang didasarkan pada medan dengan struktur tambahan. Teorema dasar dalam analisis bergantung pada sifat struktural medan bilangan riil. Yang terpenting untuk tujuan aljabar, medan yang digunakan sebagai skalar untuk ruang vektor, yang merupakan konteks umum standar untuk aljabar linear. bagian dari medan bilangan rasional, dipelajari secara mendalam di teori bilangan. dapat membantu mendeskripsikan sifat objek geometris. (in)</li> <li style="display:none;">En mathématiques, un corps commutatif (parfois simplement appelé corps, voir plus bas, ou parfois appelé champ) est une des structures algébriques fondamentales de l'algèbre générale. C'est un ensemble muni de deux opérations binaires rendant possibles les additions, soustractions, multiplications et divisions. Plus précisément, un corps commutatif est un anneau commutatif dans lequel l'ensemble des éléments non nuls est un groupe commutatif pour la multiplication. Selon la définition choisie d'un corps qui diffère selon les auteurs (la commutativité de la multiplication n'est pas toujours imposée), soit les corps commutatifs sont des cas particuliers de corps (dans le cas où la commutativité n'est pas imposée), soit la dénomination corps commutatif est un pléonasme qui désigne simplement un corps (dans le cas où elle l'est). On renvoie à l'article corps (mathématiques) pour plus de détails. Des exemples élémentaires de corps commutatifs sont le corps des nombres rationnels noté ℚ (ou Q), le corps des nombres réels noté ℝ (ou R), le corps des nombres complexes noté ℂ (ou C) et le corps ℤ/pℤ des classes de congruences modulo p où p est un nombre premier, noté alors également 𝔽p (ou Fp). La théorie des corps commutatifs est le cadre historique de la théorie de Galois, une méthode d'étude qui s'applique en particulier aux corps commutatifs et aux extensions de corps, en relation avec la théorie des groupes, mais s'étend aussi à d'autres domaines, par exemple l'étude des équations différentielles (théorie de Galois différentielle), ou des revêtements. (fr)</li> <li style="display:none;">抽象代数学において可換体（かかんたい、仏: corps commutatif）あるいは単に体（たい、英: field）とは、零でない可換可除環、あるいは同じことだが非零元全体が乗法の下で可換群をなすような環のことである。そのようなものとして体は、適当なアーベル群の公理とを満たすような加法、減法、乗法、除法の概念を備えた代数的構造である。最もよく使われる体は、実数体、複素数体、有理数体であるが、他にも有限体、関数の体、代数体、p 進数体、などがある。任意の体は、線型代数の標準的かつ一般的な対象であるベクトル空間のスカラーとして使うことができる。（ガロワ理論を含む）体拡大の理論は、ある体に係数を持つ多項式の根に関係する。他の結果として、この理論により、古典的な問題である定規とコンパスを用いた角の三等分問題や円積問題が不可能であることの証明や五次方程式が代数的に解けないというアーベル・ルフィニの定理の証明が得られる。現代数学において、体論は数論や代数幾何において必要不可欠な役割を果たしている。代数的構造として、すべての体は環であるが、すべての環が体であるわけではない。最も重要な違いは、体は（ゼロ除算を除いて）除算ができるが、環は乗法逆元がなくてもよいということである。例えば、整数の全体は環をなすが、2x = 1 は整数において解を持たない。また、体における乗法演算は可換でなければならない。可換性を仮定しない除法の可能な環は可除環、斜体、あるいは体と呼ばれる。環として、体は整域の特別なタイプとして分類でき、以下のようなクラスの包含の鎖がある。可換環 ⊃ 整域 ⊃ 整閉整域 ⊃ 一意分解整域 ⊃ 主イデアル整域 ⊃ ユークリッド整域 ⊃ ⊃ 有限体体をアルファベットで表すときは、K （続いて L, M 等）を用いる慣例がある。これは体がドイツ語で "Körper" だからである。英語の "field" の頭文字をとって F が用いられることもある。F の次の文字 G は群と紛らわしいから、前の文字 E も用いられる。 (ja)</li> <li style="display:none;">In matematica, un campo è una struttura algebrica composta da un insieme non vuoto e da due operazioni binarie interne (chiamate somma e prodotto e indicate di solito rispettivamente con e ) che godono di proprietà assimilabili a quelle verificate da somma e prodotto sui numeri razionali o reali o anche complessi. Il campo è una struttura algebrica basilare in matematica, necessaria per lo studio approfondito dei polinomi e delle loro radici, e per la definizione degli spazi vettoriali. Nel contesto degli spazi vettoriali un elemento di un campo è detto scalare. (it)</li> <li style="display:none;">체(體, 독일어: Körper, 프랑스어: corps, 영어: field)는 추상대수학에서 사칙연산이 자유로이 시행될 수 있고 산술의 잘 알려진 규칙들을 만족하는 대수 구조이다. 모든 체는 가환환이지만, 그 역은 성립하지 않는다. 체를 연구하는 추상대수학의 분야를 체론(體論, 독일어: Körpertheorie, 프랑스어: théorie des corps,영어: field theory)이라고 한다. (ko)</li> <li style="display:none;">Een lichaam (Nederlands-Nederlandse term) of veld (Belgisch-Nederlandse term), niet te verwarren met het ruimere begrip delingsring (Ned) / lichaam (Be), is een algebraïsche structuur waarin de bewerkingen optellen, aftrekken, vermenigvuldigen en delen op de gebruikelijke wijze uitgevoerd kunnen worden. In het Engelse taalgebied spreekt men van 'field', en in het Duitse taalgebied van 'Körper'. De rationale getallen, de reële getallen en de complexe getallen zijn voorbeelden van lichamen, alle met oneindig veel elementen. Is het aantal elementen van het lichaam eindig, dan spreekt men van een eindig lichaam/veld. (nl)</li> <li style="display:none;">Em matemática, um corpo é um anel comutativo com unidade em que todo elemento diferente de 0 possui um elemento inverso com relação à multiplicação. (pt)</li> <li style="display:none;">По́ле в общей алгебре — множество, для элементов которого определены операции сложения, взятия противоположного значения, умножения и деления (кроме деления на ноль), причём свойства этих операций близки к свойствам обычных числовых операций. Простейшим полем является поле рациональных чисел (дробей). Элементы поля не обязательно являются числами, поэтому, несмотря на то, что названия операций поля взяты из арифметики, определения операций могут быть далеки от арифметических. Поле — основной предмет изучения теории полей. Рациональные, вещественные, комплексные числа, рациональные функции и вычеты по модулю заданного простого числа образуют поля. (ru)</li> <li style="display:none;">Ciało – struktura formalizująca własności algebraiczne liczb wymiernych czy liczb rzeczywistych. W trakcie badań nad tymi obiektami rozwinął się aparat matematyczny (tzw. teoria Galois) umożliwiający rozwiązanie takich problemów jak rozwiązalność równań wielomianowych (jednej zmiennej) przez tzw. pierwiastniki (działania obowiązujące w ciałach i wyciąganie pierwiastków) czy wykonalność pewnych konstrukcji klasycznych (konstrukcji geometrycznych, w których dozwolone jest korzystanie z wyidealizowanych cyrkla i linijki). Działem matematyki zajmującym się opisem tych struktur jest teoria ciał. (pl)</li> <li style="display:none;">Inom högre algebra är en kropp (en. field, ty. Körper) en typ av algebraisk struktur vars egenskaper liknar dem, som till exempel de komplexa och reella talen besitter med operationerna addition och multiplikation. (sv)</li> <li style="display:none;">По́ле (англ. field — поле, нім. körper — тіло) — алгебрична структура, для якої визначено дві пари бінарних операцій: додавання/віднімання та множення/ділення, що задовольняють умовам, подібним до властивостей арифметичних операцій над раціональними, дійсними або комплексними числами. (uk)</li> <li style="display:none;">在抽象代数中，體（德語：Körper，英語：Field）是一种集合，在這個集合中可以對集合的非零元素進行加減乘除，其運算的定義與行為就如同有理數還有實數一樣。體的概念是数域以及四则运算的推廣。因此體是一個廣泛運用在代數、數論還有其他數學領域中的代數結構。體是環的一種。域和一般的環的區別在於域要求它的非零元素可以進行除法運算，這等於說每個非零的元素都要有乘法逆元。體中的運算關於乘法是可交換的。若乘法運算沒有要求可交換則稱為除環（division ring或skew field )。最有名的體結構的例子就是有理數體、實數體還有複數體。還有其他形式的體，例如有理函數體、代數函數體、代數數體、p進數體等，都很常在數學的領域中被使用或是研究，特別是數論或是代數幾何。此外還有一些密碼學上的安全協定都是依靠著有限體。在兩個體中的關係被表示成體擴張的觀念。Galois理論，由ÉvaristeGalois在1830年代提出，致力於理解體擴展的對稱性。其中Galois理論還有其他結果，解決了不能用尺規作圖做出三等份角以及化方為圓的問題。此外，還解決了五次方程不能有公式解的問題。 (zh)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/thumbnail">dbo:thumbnail</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="dbo:thumbnail" resource="http://commons.wikimedia.org/wiki/Special:FilePath/Regular_polygon_7_annotated.svg?width=300" 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resource="http://dbpedia.org/resource/Determinant_(mathematics)" href="http://dbpedia.org/resource/Determinant_(mathematics)">dbr:Determinant_(mathematics)</a></li> <li><a class="uri" rel="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Imaginary_quadratic_field" href="http://dbpedia.org/resource/Imaginary_quadratic_field">dbr:Imaginary_quadratic_field</a></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/author">dbp:author</a> </td><td class="col-10 text-break"><ul> <li>Richard Dedekind, 1871 (en)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/id">dbp:id</a> </td><td class="col-10 text-break"><ul> <li>p/f040090 (en)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/text">dbp:text</a> </td><td class="col-10 text-break"><ul> <li>By a field we will mean every infinite system of real or complex numbers so closed in itself and perfect that addition, subtraction, multiplication, and division of any two of these numbers again yields a number of the system. (en)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/title">dbp:title</a> </td><td class="col-10 text-break"><ul> <li>Field (en)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/wikiPageUsesTemplate">dbp:wikiPageUsesTemplate</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Springer" href="http://dbpedia.org/resource/Template:Springer">dbt:Springer</a></li> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:!" href="http://dbpedia.org/resource/Template:!">dbt:!</a></li> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:About" href="http://dbpedia.org/resource/Template:About">dbt:About</a></li> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" 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href="http://dbpedia.org/ontology/Building">dbo:Building</a></li> <li><a class="uri" rel="rdf:type" resource="http://dbpedia.org/class/yago/Structure104341686" href="http://dbpedia.org/class/yago/Structure104341686">yago:Structure104341686</a></li> <li><a class="uri" rel="rdf:type" resource="http://dbpedia.org/class/yago/Whole100003553" href="http://dbpedia.org/class/yago/Whole100003553">yago:Whole100003553</a></li> <li><a class="uri" rel="rdf:type" resource="http://dbpedia.org/class/yago/WikicatAlgebraicStructures" href="http://dbpedia.org/class/yago/WikicatAlgebraicStructures">yago:WikicatAlgebraicStructures</a></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#comment">rdfs:comment</a> </td><td class="col-10 text-break"><ul> <li style="display:none;">En l'àlgebra abstracta, un cos és un sistema algebraic en què és possible efectuar la suma, resta, multiplicació i divisió (llevat de la divisió per 0), i en la qual se satisfan certes lleis. (ca)</li> <li style="display:none;">Komutativní těleso (někdy stručně těleso podle německého körper, někdy též pole z anglického field) je algebraická struktura, na které jsou definovány dvě binární operace, sčítání a násobení, pro které platí řada určených vlastností. Jedná se o taková tělesa, kde násobení splňuje navíc komutativitu, respektive takové , kde navíc existuje inverzní prvek pro obě binární operace (okruh vyžadoval existenci inverzního prvku jen pro operaci +). Tělesa, ve kterých násobení není komutativní, se nazývají . (cs)</li> <li style="display:none;">En matematiko kaj, pli specife, en abstrakta algebro kampo estas komuta korpo. Tio estas unu el la plej gravaj nocioj de multaj fakoj de abstrakta algebro kaj nombro-teorio. Pli detale, oni povas karakterizi la nocion kampo K per ĉi-subaj aksiomoj. (eo)</li> <li style="display:none;">Ein Körper ist im mathematischen Teilgebiet der Algebra eine ausgezeichnete algebraische Struktur, in der die Addition, Subtraktion, Multiplikation und Division auf eine bestimmte Weise durchgeführt werden können. Die Bezeichnung „Körper“ wurde im 19. Jahrhundert von Richard Dedekind eingeführt. Die wichtigsten Körper, die in fast allen Gebieten der Mathematik benutzt werden, sind der Körper der rationalen Zahlen, der Körper der reellen Zahlen und der Körper der komplexen Zahlen. (de)</li> <li style="display:none;">Aljebra abstraktuan gorputza da multzoak gutxienez bi elementu izanda, multzorako (gehiketa) eragiketak elkartze eta trukatze propietatea eta elementu alderantzizko eta neutroaren existentzia betetzen dituen, eta (biderketa edo produktua)eragiketak elkartze eta banatze propietateak, elementu neutroaren existentzia eta multzoko elementuentzako, gehiketarekiko elementu neutroa izan ezik, elementu alderantzizkoaren existentzia betetzen dituen egitura aljebraikoa. (eu)</li> <li style="display:none;">In matematica, un campo è una struttura algebrica composta da un insieme non vuoto e da due operazioni binarie interne (chiamate somma e prodotto e indicate di solito rispettivamente con e ) che godono di proprietà assimilabili a quelle verificate da somma e prodotto sui numeri razionali o reali o anche complessi. Il campo è una struttura algebrica basilare in matematica, necessaria per lo studio approfondito dei polinomi e delle loro radici, e per la definizione degli spazi vettoriali. Nel contesto degli spazi vettoriali un elemento di un campo è detto scalare. (it)</li> <li style="display:none;">체(體, 독일어: Körper, 프랑스어: corps, 영어: field)는 추상대수학에서 사칙연산이 자유로이 시행될 수 있고 산술의 잘 알려진 규칙들을 만족하는 대수 구조이다. 모든 체는 가환환이지만, 그 역은 성립하지 않는다. 체를 연구하는 추상대수학의 분야를 체론(體論, 독일어: Körpertheorie, 프랑스어: théorie des corps,영어: field theory)이라고 한다. (ko)</li> <li style="display:none;">Een lichaam (Nederlands-Nederlandse term) of veld (Belgisch-Nederlandse term), niet te verwarren met het ruimere begrip delingsring (Ned) / lichaam (Be), is een algebraïsche structuur waarin de bewerkingen optellen, aftrekken, vermenigvuldigen en delen op de gebruikelijke wijze uitgevoerd kunnen worden. In het Engelse taalgebied spreekt men van 'field', en in het Duitse taalgebied van 'Körper'. De rationale getallen, de reële getallen en de complexe getallen zijn voorbeelden van lichamen, alle met oneindig veel elementen. Is het aantal elementen van het lichaam eindig, dan spreekt men van een eindig lichaam/veld. (nl)</li> <li style="display:none;">Em matemática, um corpo é um anel comutativo com unidade em que todo elemento diferente de 0 possui um elemento inverso com relação à multiplicação. (pt)</li> <li style="display:none;">Ciało – struktura formalizująca własności algebraiczne liczb wymiernych czy liczb rzeczywistych. W trakcie badań nad tymi obiektami rozwinął się aparat matematyczny (tzw. teoria Galois) umożliwiający rozwiązanie takich problemów jak rozwiązalność równań wielomianowych (jednej zmiennej) przez tzw. pierwiastniki (działania obowiązujące w ciałach i wyciąganie pierwiastków) czy wykonalność pewnych konstrukcji klasycznych (konstrukcji geometrycznych, w których dozwolone jest korzystanie z wyidealizowanych cyrkla i linijki). Działem matematyki zajmującym się opisem tych struktur jest teoria ciał. (pl)</li> <li style="display:none;">Inom högre algebra är en kropp (en. field, ty. Körper) en typ av algebraisk struktur vars egenskaper liknar dem, som till exempel de komplexa och reella talen besitter med operationerna addition och multiplikation. (sv)</li> <li style="display:none;">По́ле (англ. field — поле, нім. körper — тіло) — алгебрична структура, для якої визначено дві пари бінарних операцій: додавання/віднімання та множення/ділення, що задовольняють умовам, подібним до властивостей арифметичних операцій над раціональними, дійсними або комплексними числами. (uk)</li> <li style="display:none;">在抽象代数中，體（德語：Körper，英語：Field）是一种集合，在這個集合中可以對集合的非零元素進行加減乘除，其運算的定義與行為就如同有理數還有實數一樣。體的概念是数域以及四则运算的推廣。因此體是一個廣泛運用在代數、數論還有其他數學領域中的代數結構。體是環的一種。域和一般的環的區別在於域要求它的非零元素可以進行除法運算，這等於說每個非零的元素都要有乘法逆元。體中的運算關於乘法是可交換的。若乘法運算沒有要求可交換則稱為除環（division ring或skew field )。最有名的體結構的例子就是有理數體、實數體還有複數體。還有其他形式的體，例如有理函數體、代數函數體、代數數體、p進數體等，都很常在數學的領域中被使用或是研究，特別是數論或是代數幾何。此外還有一些密碼學上的安全協定都是依靠著有限體。在兩個體中的關係被表示成體擴張的觀念。Galois理論，由ÉvaristeGalois在1830年代提出，致力於理解體擴展的對稱性。其中Galois理論還有其他結果，解決了不能用尺規作圖做出三等份角以及化方為圓的問題。此外，還解決了五次方程不能有公式解的問題。 (zh)</li> <li style="display:none;">في الرياضيات، الحقل (بالإنجليزية: Field)‏ هي مجموعة عُرفت عليها أربع عمليات هن الجمع والطرح والضرب والقسمة ويحقق خواص ما يقابلها من عمليات الأعداد الحقيقة والنسبية. فبالتالي الحقل هو بنية أساسية في الجبر ويستخدم بشكل واسع في الجبر ونظرية الأعداد وغيره من فروع الرياضيات. (ar)</li> <li style="display:none;">Σώμα (από το γαλλικό Corps) είναι ένα σύνολο (από το αγγλικό Field) αντικειμένων οποιουδήποτε είδους, μαζί με δύο δυαδικές πράξεις + και * ορισμένες στο , οι οποίες απεικονίζουν 2 στοιχεία a και b που ανήκουν στο F στα a+b και a*b, επίσης στοιχεία του F.Και ισχύουν οι εξής ιδιότητες: Εκτός από τα γνωστά παραδείγματα σωμάτων υπάρχουν και τα παραδείγματα των σωμάτων που είναι της μορφής a+b* και γενικά της μορφής αυτής που το υπόρριζο μπορεί να πάρει τις τιμές 2,3,...,ν. Ένας δακτύλιος καλείται σώμα αν ισχύουν τα εξής : (el)</li> <li>In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. (en)</li> <li style="display:none;">En matemática, concretamente en el campo de la álgebra abstracta, un cuerpo (en ocasiones llamado campo como traducción de inglés field) es un sistema algebraico en el cual las operaciones llamadas adición y multiplicación se pueden realizar y cumplen las propiedades: asociativa, conmutativa y distributiva de la multiplicación respecto de la adición, además de la existencia de inverso aditivo, de inverso multiplicativo y de un elemento neutro para la adición y otro para la multiplicación, los cuales permiten efectuar las operaciones de sustracción y división (excepto la división por cero); estas propiedades ya son familiares de la aritmética de números racionales. (es)</li> <li style="display:none;">En mathématiques, un corps commutatif (parfois simplement appelé corps, voir plus bas, ou parfois appelé champ) est une des structures algébriques fondamentales de l'algèbre générale. C'est un ensemble muni de deux opérations binaires rendant possibles les additions, soustractions, multiplications et divisions. Plus précisément, un corps commutatif est un anneau commutatif dans lequel l'ensemble des éléments non nuls est un groupe commutatif pour la multiplication. (fr)</li> <li style="display:none;">Lapangan atau medan (juga disebut bidang) dalam matematika adalah suatu struktur aljabar dengan operasi seperti penambahan, pengurangan, perkalian, dan pembagian yang memenuhi aksioma tertentu. Lapangan yang kerap kali dijumpai adalah lapangan bilangan riil, lapangan bilangan kompleks dan bilangan rasional. (in)</li> <li style="display:none;">抽象代数学において可換体（かかんたい、仏: corps commutatif）あるいは単に体（たい、英: field）とは、零でない可換可除環、あるいは同じことだが非零元全体が乗法の下で可換群をなすような環のことである。そのようなものとして体は、適当なアーベル群の公理とを満たすような加法、減法、乗法、除法の概念を備えた代数的構造である。最もよく使われる体は、実数体、複素数体、有理数体であるが、他にも有限体、関数の体、代数体、p 進数体、などがある。任意の体は、線型代数の標準的かつ一般的な対象であるベクトル空間のスカラーとして使うことができる。（ガロワ理論を含む）体拡大の理論は、ある体に係数を持つ多項式の根に関係する。他の結果として、この理論により、古典的な問題である定規とコンパスを用いた角の三等分問題や円積問題が不可能であることの証明や五次方程式が代数的に解けないというアーベル・ルフィニの定理の証明が得られる。現代数学において、体論は数論や代数幾何において必要不可欠な役割を果たしている。環として、体は整域の特別なタイプとして分類でき、以下のようなクラスの包含の鎖がある。可換環 ⊃ 整域 ⊃ 整閉整域 ⊃ 一意分解整域 ⊃ 主イデアル整域 ⊃ ユークリッド整域 ⊃ ⊃ 有限体 (ja)</li> <li style="display:none;">По́ле в общей алгебре — множество, для элементов которого определены операции сложения, взятия противоположного значения, умножения и деления (кроме деления на ноль), причём свойства этих операций близки к свойствам обычных числовых операций. Простейшим полем является поле рациональных чисел (дробей). Элементы поля не обязательно являются числами, поэтому, несмотря на то, что названия операций поля взяты из арифметики, определения операций могут быть далеки от арифметических. (ru)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#label">rdfs:label</a> </td><td class="col-10 text-break"><ul> <li>Field (mathematics) (en)</li> <li style="display:none;">حقل (رياضيات) (ar)</li> <li style="display:none;">Cos (matemàtiques) (ca)</li> <li style="display:none;">Komutativní těleso (cs)</li> <li style="display:none;">Körper (Algebra) (de)</li> <li style="display:none;">Σώμα (άλγεβρα) (el)</li> <li style="display:none;">Kampo (algebro) (eo)</li> <li style="display:none;">Cuerpo (matemáticas) (es)</li> <li style="display:none;">Gorputz (matematika) (eu)</li> <li style="display:none;">Medan (matematika) (in)</li> <li style="display:none;">Campo (matematica) (it)</li> <li style="display:none;">Corps commutatif (fr)</li> <li style="display:none;">체 (수학) (ko)</li> <li style="display:none;">可換体 (ja)</li> <li style="display:none;">Lichaam (Ned) / Veld (Be) (nl)</li> <li style="display:none;">Ciało (matematyka) (pl)</li> <li style="display:none;">Corpo (matemática) (pt)</li> <li style="display:none;">Поле (алгебра) (ru)</li> <li style="display:none;">Kropp (algebra) (sv)</li> <li style="display:none;">域 (数学) (zh)</li> <li style="display:none;">Поле (алгебра) (uk)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2002/07/owl#sameAs">owl:sameAs</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="owl:sameAs" resource="http://rdf.freebase.com/ns/m.02w23" href="http://rdf.freebase.com/ns/m.02w23">freebase:Field (mathematics)</a></li> <li><a class="uri" rel="owl:sameAs" resource="http://yago-knowledge.org/resource/Field_(mathematics)" href="http://yago-knowledge.org/resource/Field_(mathematics)">yago-res:Field (mathematics)</a></li> <li><a class="uri" rel="owl:sameAs" resource="http://www.wikidata.org/entity/Q190109" href="http://www.wikidata.org/entity/Q190109">wikidata:Field (mathematics)</a></li> <li><a class="uri" rel="owl:sameAs" 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