Teorija polja (matematika)

Regularni heptagon se ne može konstruisati koristeći samo konstrukcije lenjirom i šestarom; to se može dokazati koristeći polje konstruktivnih brojeva.

U matematici, polje je skup na kome su sabiranje, oduzimanje, množenje, i deljenje definisani, i ponašaju se kao korespondirajuće operacije na racionalnim i realnim brojevima. Polje je stoga fundamentalna algebarska struktura, koja ima široku primenu u algebri, teoriji brojeva i mnogim drugim oblastima matematike.

Najbolje poznata polja su polje racionalnih brojeva, polje 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 algebraically unsolvable.

Fields serve as foundational notions in several mathematical domains. This includes different branches of 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.

Definicija

Informally, a field is a set, along with two operations defined on that set: an addition operation written as a + b, and a multiplication operation written as a ⋅ b, both of which behave similarly as they behave for rational numbers and real numbers, including the existence of an additive inverse −a for all elements a, and of a multiplicative inverse b⁻¹ for every nonzero element b. This allows one to also consider the so-called inverse operations of subtraction, a − b, and division, a / b, by defining:

a − b = a + (−b),

a / b = a · b⁻¹.

Klasična definicija

Formally, a field is a set F together with two operations on F called addition and multiplication. An operation on F is a function F × F → F – in other words, a mapping that associates an element of F to every pair of its elements. The result of the addition of a and b is called the sum of a and b, and is denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, and is denoted ab or a ⋅ b. These operations are required to satisfy the following properties, referred to as field axioms. In these axioms, a, b, and c are arbitrary elements of the field F.

• Associativity of addition and multiplication: a + (b + c) = (a + b) + c, and a · (b · c) = (a · b) · c.
• Commutativity of addition and multiplication: a + b = b + a, and a · b = b · a.
• Additive and multiplicative identity: there exist two different elements 0 and 1 in F such that a + 0 = a and a · 1 = a.
• Additive inverses: for every a in F, there exists an element in F, denoted −a, called the additive inverse of a, such that a + (−a) = 0.
• Multiplicative inverses: for every a ≠ 0 in F, there exists an element in F, denoted by a⁻¹ or 1/a, called the multiplicative inverse of a, such that a · a⁻¹ = 1.
• Distributivity of multiplication over addition: a · (b + c) = (a · b) + (a · c).

This may be summarized by saying: a field has two operations, called addition and multiplication; it is an abelian group under addition with 0 as the additive identity; the nonzero elements are an abelian group under multiplication with 1 as the multiplicative identity; and multiplication distributes over addition.

Literatura

• Adamson, I. T. (2007), Introduction to Field Theory, Dover Publications, ISBN 978-0-486-46266-0 
• Allenby, R. B. J. T. (1991), Rings, Fields and Groups, Butterworth-Heinemann, ISBN 978-0-340-54440-2 
• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-13-004763-2 , especially Chapter 13
• Artin, Emil; Schreier, Otto (1927), „Eine Kennzeichnung der reell abgeschlossenen Körper”, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (на језику: German), 5: 225—231, ISSN 0025-5858, JFM 53.0144.01, doi:10.1007/BF02952522 CS1 одржавање: Непрепознат језик (веза)
• Ax, James (1968), „The elementary theory of finite fields”, Ann. of Math., 2, 88: 239—271, doi:10.2307/1970573 
• Baez, John C. (2002), „The octonions”, Bulletin of the American Mathematical Society, 39: 145—205, arXiv:math/0105155 , doi:10.1090/S0273-0979-01-00934-X 
• Banaschewski, Bernhard (1992), „Algebraic closure without choice.”, Z. Math. Logik Grundlagen Math., 38 (4): 383—385, Zbl 0739.03027 
• Beachy, John. A; Blair, William D. (2006), Abstract Algebra (3 изд.), Waveland Press, ISBN 1-57766-443-4 
• Blyth, T. S.; Robertson, E. F. (1985), Groups, rings and fields: Algebra through practice, Cambridge University Press . See especially Book 3 (ISBN 0-521-27288-2) and Book 6 (ISBN 0-521-27291-2).
• Borceux, Francis; Janelidze, George (2001), Galois theories, Cambridge University Press, ISBN 0-521-80309-8, Zbl 0978.12004 
• Bourbaki, Nicolas (1994), Elements of the history of mathematics, Springer, ISBN 3-540-19376-6, MR 1290116, doi:10.1007/978-3-642-61693-8 
• Bourbaki, Nicolas (1988), Algebra II. Chapters 4–7, Springer, ISBN 0-387-19375-8 
• Cassels, J. W. S. (1986), Local fields, London Mathematical Society Student Texts, 3, Cambridge University Press, ISBN 0-521-30484-9, MR 861410, doi:10.1017/CBO9781139171885 
• Clark, A. (1984), Elements of Abstract Algebra, Dover Books on Mathematics Series, Dover, ISBN 978-0-486-64725-8 
• Conway, John Horton (1976), On Numbers and Games, Academic Press 
• Corry, Leo (2004), Modern algebra and the rise of mathematical structures (2nd изд.), Birkhäuser, ISBN 3-7643-7002-5, Zbl 1044.01008 
• Dirichlet, Peter Gustav Lejeune (1871), Dedekind, Richard, ур., Vorlesungen über Zahlentheorie (Lectures on Number Theory) (на језику: German), 1 (2nd изд.), Braunschweig, Germany: Friedrich Vieweg und Sohn CS1 одржавање: Непрепознат језик (веза)
• Eisenbud, David (1995), Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, 150, New York: Springer-Verlag, ISBN 0-387-94268-8, MR 1322960, doi:10.1007/978-1-4612-5350-1 
• Escofier, J. P. (2012), Galois Theory, Springer, ISBN 978-1-4613-0191-2 
• Fricke, Robert; Weber, Heinrich Martin (1924), Lehrbuch der Algebra (на језику: German), Vieweg, JFM 50.0042.03 CS1 одржавање: Непрепознат језик (веза)
• Gouvêa, Fernando Q. (1997), p-adic numbers, Universitext (2nd изд.), Springer 
• Gouvêa, Fernando Q. (2012), A Guide to Groups, Rings, and Fields, Mathematical Association of America, ISBN 978-0-88385-355-9 
• Hazewinkel Michiel, ур. (2001). „Field”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104. 
• Hensel, Kurt (1904), „Über eine neue Begründung der Theorie der algebraischen Zahlen”, Journal für die Reine und Angewandte Mathematik (на језику: German), 128: 1—32, ISSN 0075-4102, JFM 35.0227.01 CS1 одржавање: Непрепознат језик (веза)
• Jacobson, Nathan (2009), Basic algebra, 1 (2nd изд.), Dover, ISBN 978-0-486-47189-1 
• Jannsen, Uwe; Wingberg, Kay (1982), „Die Struktur der absoluten Galoisgruppe 𝔭-adischer Zahlkörper. [The structure of the absolute Galois group of 𝔭-adic number fields]”, Invent. Math., 70 (1): 71—98, Bibcode:1982InMat..70...71J, MR 0679774, doi:10.1007/bf01393199 
• Kleiner, Israel (2007), A history of abstract algebra, Birkhäuser, ISBN 978-0-8176-4684-4, MR 2347309, doi:10.1007/978-0-8176-4685-1 
• Kiernan, B. Melvin (1971), „The development of Galois theory from Lagrange to Artin”, Archive for History of Exact Sciences, 8 (1-2): 40—154, MR 1554154, doi:10.1007/BF00327219 
• Kuhlmann, Salma (2000), Ordered exponential fields, Fields Institute Monographs, 12, American Mathematical Society, ISBN 0-8218-0943-1, MR 1760173 
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (3rd изд.), Springer, ISBN 0-387-95385-X, doi:10.1007/978-1-4613-0041-0 
• Lidl, Rudolf; Niederreiter, Harald (2008), Finite fields (2nd изд.), Cambridge University Press, ISBN 978-0-521-06567-2, Zbl 1139.11053 
• Lorenz, Falko (2008), Algebra, Volume II: Fields with Structures, Algebras and Advanced Topics, Springer, ISBN 978-0-387-72487-4 
• Marker, David; Messmer, Margit; Pillay, Anand (2006), Model theory of fields, Lecture Notes in Logic, 5 (2nd изд.), Association for Symbolic Logic, CiteSeerX 10.1.1.36.8448 , ISBN 978-1-56881-282-3, MR 2215060 
• Mines, Ray; Richman, Fred; Ruitenburg, Wim (1988), A course in constructive algebra, Universitext, Springer, ISBN 0-387-96640-4, MR 919949, doi:10.1007/978-1-4419-8640-5 
• Moore, E. Hastings (1893), „A doubly-infinite system of simple groups”, Bulletin of the American Mathematical Society, 3 (3): 73—78, MR 1557275, doi:10.1090/S0002-9904-1893-00178-X 
• Prestel, Alexander (1984), Lectures on formally real fields, Lecture Notes in Mathematics, 1093, Springer, ISBN 3-540-13885-4, MR 769847, doi:10.1007/BFb0101548 
• Ribenboim, Paulo (1999), The theory of classical valuations, Springer Monographs in Mathematics, Springer, ISBN 0-387-98525-5, MR 1677964, doi:10.1007/978-1-4612-0551-7 
• Scholze, Peter (2014), „Perfectoid spaces and their Applications” (PDF), Proceedings of the International Congress of Mathematicians 2014, ISBN 978-89-6105-804-9 
• Schoutens, Hans (2002), The Use of Ultraproducts in Commutative Algebra, Lecture Notes in Mathematics, 1999, Springer, ISBN 978-3-642-13367-1 
• Serre, Jean-Pierre (1996) [1978], A course in arithmetic. Translation of Cours d'arithmetique, Graduate Text in Mathematics, 7 (2nd изд.), Springer, ISBN 9780387900407, Zbl 0432.10001 
• Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics, 67, Springer, ISBN 0-387-90424-7, MR 554237 
• Serre, Jean-Pierre (1992), Topics in Galois theory, Jones and Bartlett Publishers, ISBN 0-86720-210-6, Zbl 0746.12001 
• Serre, Jean-Pierre (2002), Galois cohomology, Springer Monographs in Mathematics, Translated from the French by Patrick Ion, Berlin, New York: Springer-Verlag, ISBN 978-3-540-42192-4, MR 1867431, Zbl 1004.12003 
• Sharpe, David (1987), Rings and factorization, Cambridge University Press, ISBN 0-521-33718-6, Zbl 0674.13008 
• Steinitz, Ernst (1910), „Algebraische Theorie der Körper” [Algebraic Theory of Fields], Journal für die reine und angewandte Mathematik, 137: 167—309, ISSN 0075-4102, JFM 41.0445.03, doi:10.1515/crll.1910.137.167 
• Tits, Jacques (1957), „Sur les analogues algébriques des groupes semi-simples complexes”, Colloque d'algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques Établissements Ceuterick, Louvain, Paris: Librairie Gauthier-Villars, стр. 261—289 
• van der Put, M.; Singer, M. F. (2003), Galois Theory of Linear Differential Equations, Grundlehren der mathematischen Wissenschaften, 328, Springer 
• von Staudt, Karl Georg Christian (1857), Beiträge zur Geometrie der Lage (Contributions to the Geometry of Position), 2, Nürnberg (Germany): Bauer and Raspe 
• Wallace, D. A. R. (1998), Groups, Rings, and Fields, SUMS, 151, Springer 
• Warner, Seth (1989), Topological fields, North-Holland, ISBN 0-444-87429-1, Zbl 0683.12014 
• Washington, Lawrence C. (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, 83 (2nd изд.), Springer-Verlag, ISBN 0-387-94762-0, MR 1421575, doi:10.1007/978-1-4612-1934-7 
• Weber, Heinrich (1893), „Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie”, Mathematische Annalen (на језику: German), 43: 521—549, ISSN 0025-5831, JFM 25.0137.01, doi:10.1007/BF01446451 CS1 одржавање: Непрепознат језик (веза)