Поље алгебарских бројева

У апстрактној алгебри поље алгебарских бројева се означава са F и представља коначно проширење поља рационалних бројева Q, то јест, поље које садржи поље рационалних бројева и има коначну димензију, када се посматра као векторски простор над Q. Ова поља су врло важна у теорији бројева и представљају центар студија која се баве теоријом алгебарских бројева.

Појам се ослања на сам концепт поља у математици, које представља алгебарску структуру сачињену од скупа елемената и две операције дефинисане на том скупу. Те операције се називају сабирање и множење и да би чиниле поље морају имати својство дистрибутивности.

Концепт поља је увео Дедекинд, који је користио немачку реч Körper (тело) за овај појам. Најједноставнији пример је управо поље рационалних бројева Q. Поље реалних бројева R и поље комплексних бројева C су такође примери поља алгебарских бројева.

Дефиниција

Предуслови

Главни чланци: Field и Vector space

The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted $\mathbb {Q}$, together with its usual operations of addition and multiplication.

Another notion needed to define algebraic number fields is vector spaces. To the extent needed here, vector spaces can be thought of as consisting of sequences (or tuples)

(x₁, x₂, …)

whose entries are elements of a fixed field, such as the field $\mathbb {Q}$. Any two such sequences can be added by adding the entries one per one. Furthermore, any sequence can be multiplied by a single element c of the fixed field. These two operations known as vector addition and scalar multiplication satisfy a number of properties that serve to define vector spaces abstractly. Vector spaces are allowed to be "infinite-dimensional", that is to say that the sequences constituting the vector spaces are of infinite length. If, however, the vector space consists of finite sequences

(x₁, x₂, …, x_n),

the vector space is said to be of finite dimension, n.

Дефиниција

An algebraic number field (or simply number field) is a finite-degree field extension of the field of rational numbers. Here degree means the dimension of the field as a vector space over $\mathbb {Q}$.

Примери

• The smallest and most basic number field is the field $\mathbb {Q}$ of rational numbers. Many properties of general number fields are modeled after the properties of $\mathbb {Q}$.
• The Gaussian rationals, denoted $\mathbb {Q} (i)$ (read as "$\mathbb {Q}$ adjoined $i$"), form the first nontrivial example of a number field. Its elements are expressions of the form

  $a+bi$

  where both a and b are rational numbers and i is the imaginary unit. Such expressions may be added, subtracted, and multiplied according to the usual rules of arithmetic and then simplified using the identity

  $i^{2}=-1$.

  Explicitly,

  ${\begin{matrix}(a+bi)+(c+di)&=&(a+c)+(b+d)i\\(a+bi)\cdot (c+di)&=&(ac-bd)+(ad+bc)i\end{matrix}}$

  Non-zero Gaussian rational numbers are invertible, which can be seen from the identity

  $(a+bi)\left({\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}i\right)={\frac {(a+bi)(a-bi)}{a^{2}+b^{2}}}=1.$

  It follows that the Gaussian rationals form a number field which is two-dimensional as a vector space over $\mathbb {Q}$.

• More generally, for any square-free integer $d$, the quadratic field $\mathbb {Q} ({\sqrt {d}})$ is a number field obtained by adjoining the square root of $d$ to the field of rational numbers. Arithmetic operations in this field are defined in analogy with the case of Gaussian rational numbers, $d=-1$.
• Cyclotomic field

  $\mathbb {Q} (\zeta _{n})$, where $\zeta _{n}=\exp {2\pi i/n}$

  is a number field obtained from $\mathbb {Q}$ by adjoining a primitive $n$th root of unity $\zeta _{n}$. This field contains all complex nth roots of unity and its dimension over $\mathbb {Q}$ is equal to $\varphi (n)$, where $\varphi$ is the Euler totient function.

Види још

• Поље (математика)
• Апстрактна алгебра
• Алгебарска структура
• Алгебарски прстен

Литература

• Cohn, Harvey (1988), A Classical Invitation to Algebraic Numbers and Class Fields, Universitext, New York: Springer-Verlag 
• Janusz, Gerald J. (1996), Algebraic Number Fields (2nd изд.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0429-2 
• Helmut Hasse, Number Theory, Springer Classics in Mathematics Series (2002)
• Serge Lang, Algebraic Number Theory, second edition, Springer, 2000
• Richard A. Mollin, Algebraic Number Theory, CRC, 1999
• Ram Murty, Problems in Algebraic Number Theory, Second Edition, Springer, 2005
• Narkiewicz, Władysław (2004), Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics (3 изд.), Berlin: Springer-Verlag, ISBN 978-3-540-21902-6, MR 2078267 
• Neukirch, Jürgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, 322, Berlin, New York: Springer-Verlag, ISBN 978-3-540-65399-8, MR 1697859, Zbl 0956.11021 
• Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, 323, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 1136.11001 
• André Weil, Basic Number Theory, third edition, Springer, 1995
• Hazewinkel Michiel, ур. (2001). „Kummer extension”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104. 
• Bryan Birch, "Cyclotomic fields and Kummer extensions", in J.W.S. Cassels and A. Frohlich (edd), Algebraic number theory, Academic Press, 1973. Chap.III, pp. 85–93.
• Ishida, Makoto (1976). The genus fields of algebraic number fields. Lecture Notes in Mathematics. 555. Springer-Verlag. ISBN 3-540-08000-7. Zbl 0353.12001. 
• Janusz, Gerald (1973). Algebraic Number Fields. Pure and Applied Mathematics. 55. Academic Press. ISBN 0-12-380250-4. Zbl 0307.12001. 
• Lemmermeyer, Franz (2000). Reciprocity laws. From Euler to Eisenstein. Springer Monographs in Mathematics. Berlin: Springer-Verlag. ISBN 3-540-66957-4. MR 1761696. Zbl 0949.11002. 
• 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: немачки), 5: 225—231, ISSN 0025-5858, JFM 53.0144.01, S2CID 121547404, doi:10.1007/BF02952522 
• Ax, James (1968), „The elementary theory of finite fields”, Ann. of Math., 2, 88 (2): 239—271, JSTOR 1970573, doi:10.2307/1970573 
• Baez, John C. (2002), „The octonions”, Bulletin of the American Mathematical Society, 39 (2): 145—205, S2CID 586512, 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, doi:10.1002/malq.19920380136 
• 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: немачки), 1 (2nd изд.), Braunschweig, Germany: Friedrich Vieweg und Sohn 
• 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 
• Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd изд.), Reading: Addison-Wesley, ISBN 0-201-01984-1 
• Fricke, Robert; Weber, Heinrich Martin (1924), Lehrbuch der Algebra (на језику: немачки), Vieweg, JFM 50.0042.03 
• 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: немачки), 128: 1—32, ISSN 0075-4102, JFM 35.0227.01 
• 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, S2CID 119378923, doi:10.1007/bf01393199 
• Kleiner, Israel (2007), Kleiner, Israel, ур., 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, S2CID 121442989, 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 
• McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68015225 
• 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, 1910 (137): 167—309, ISSN 0075-4102, JFM 41.0445.03, S2CID 120807300, 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: немачки), 43 (4): 521—549, ISSN 0025-5831, JFM 25.0137.01, S2CID 120528969, doi:10.1007/BF01446451 

Спољашње везе

• Поља - дефиниција и основна својства.
• Dirichlet’s unit theorem, Keith Conrad