Field (mathematics) (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Field (mathematics)" in English language version.

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Some authors also consider the fields $R$ and $C$ to be local fields. On the other hand, these two fields, also called Archimedean local fields, share little similarity with the local fields considered here, to a point that Cassels (1986, p. vi) calls them "completely anomalous". Cassels, J. W. S. (1986), Local fields, London Mathematical Society Student Texts, vol. 3, Cambridge University Press, doi:10.1017/CBO9781139171885, ISBN 0-521-30484-9, MR 0861410
Mines, Richman & Ruitenburg (1988), §II.2. See also Heyting field. Mines, Ray; Richman, Fred; Ruitenburg, Wim (1988), A course in constructive algebra, Universitext, Springer, doi:10.1007/978-1-4419-8640-5, ISBN 0-387-96640-4, MR 0919949
Kleiner (2007), p. 63 Kleiner, Israel (2007), Kleiner, Israel (ed.), A history of abstract algebra, Birkhäuser, doi:10.1007/978-0-8176-4685-1, ISBN 978-0-8176-4684-4, MR 2347309
Kiernan (1971), p. 50 Kiernan, B. Melvin (1971), "The development of Galois theory from Lagrange to Artin", Archive for History of Exact Sciences, 8 (1–2): 40–154, doi:10.1007/BF00327219, MR 1554154, S2CID 121442989
Bourbaki (1994), pp. 75–76 Bourbaki, Nicolas (1994), Elements of the history of mathematics, Springer, doi:10.1007/978-3-642-61693-8, ISBN 3-540-19376-6, MR 1290116
Dirichlet (1871), p. 42, translation by Kleiner (2007), p. 66 Dirichlet, Peter Gustav Lejeune (1871), Dedekind, Richard (ed.), Vorlesungen über Zahlentheorie (Lectures on Number Theory) (in German), vol. 1 (2nd ed.), Braunschweig, Germany: Friedrich Vieweg und Sohn Kleiner, Israel (2007), Kleiner, Israel (ed.), A history of abstract algebra, Birkhäuser, doi:10.1007/978-0-8176-4685-1, ISBN 978-0-8176-4684-4, MR 2347309
Bourbaki (1994), p. 81 Bourbaki, Nicolas (1994), Elements of the history of mathematics, Springer, doi:10.1007/978-3-642-61693-8, ISBN 3-540-19376-6, MR 1290116
Bourbaki (1994), p. 92 Bourbaki, Nicolas (1994), Elements of the history of mathematics, Springer, doi:10.1007/978-3-642-61693-8, ISBN 3-540-19376-6, MR 1290116
Eisenbud (1995), p. 60 Eisenbud, David (1995), Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, New York: Springer-Verlag, doi:10.1007/978-1-4612-5350-1, ISBN 0-387-94268-8, MR 1322960
Ribenboim (1999), p. 186, §7.1 Ribenboim, Paulo (1999), The theory of classical valuations, Springer Monographs in Mathematics, Springer, doi:10.1007/978-1-4612-0551-7, ISBN 0-387-98525-5, MR 1677964
Prestel (1984), Proposition 1.22 Prestel, Alexander (1984), Lectures on formally real fields, Lecture Notes in Mathematics, vol. 1093, Springer, doi:10.1007/BFb0101548, ISBN 3-540-13885-4, MR 0769847
Prestel (1984), Theorem 1.23 Prestel, Alexander (1984), Lectures on formally real fields, Lecture Notes in Mathematics, vol. 1093, Springer, doi:10.1007/BFb0101548, ISBN 3-540-13885-4, MR 0769847
Serre (1979) Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics, vol. 67, Springer, ISBN 0-387-90424-7, MR 0554237
Marker, Messmer & Pillay (2006), Corollary 1.2 Marker, David; Messmer, Margit; Pillay, Anand (2006), Model theory of fields, Lecture Notes in Logic, vol. 5 (2nd ed.), Association for Symbolic Logic, CiteSeerX 10.1.1.36.8448, ISBN 978-1-56881-282-3, MR 2215060
Kuhlmann (2000) Kuhlmann, Salma (2000), Ordered exponential fields, Fields Institute Monographs, vol. 12, American Mathematical Society, ISBN 0-8218-0943-1, MR 1760173
Jannsen & Wingberg (1982) 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, doi:10.1007/bf01393199, MR 0679774, S2CID 119378923
Serre (2002) 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
Eisenbud (1995), §13, Theorem A Eisenbud, David (1995), Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, New York: Springer-Verlag, doi:10.1007/978-1-4612-5350-1, ISBN 0-387-94268-8, MR 1322960
Washington (1997) Washington, Lawrence C. (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83 (2nd ed.), Springer-Verlag, doi:10.1007/978-1-4612-1934-7, ISBN 0-387-94762-0, MR 1421575

archive.org

Marker, Messmer & Pillay (2006), Corollary 1.2 Marker, David; Messmer, Margit; Pillay, Anand (2006), Model theory of fields, Lecture Notes in Logic, vol. 5 (2nd ed.), Association for Symbolic Logic, CiteSeerX 10.1.1.36.8448, ISBN 978-1-56881-282-3, MR 2215060
Serre (1996), Chapter IV Serre, Jean-Pierre (1996) [1978], A course in arithmetic. Translation of Cours d'arithmetique, Graduate Text in Mathematics, vol. 7 (2nd ed.), Springer, ISBN 9780387900407, Zbl 0432.10001

arxiv.org

Baez (2002) Baez, John C. (2002), "The octonions", Bulletin of the American Mathematical Society, 39 (2): 145–205, arXiv:math/0105155, doi:10.1090/S0273-0979-01-00934-X, S2CID 586512

books.google.com

Clark (1984), Chapter 3 Clark, A. (1984), Elements of Abstract Algebra, Dover Books on Mathematics Series, Dover, ISBN 978-0-486-64725-8
Dirichlet (1871), p. 42, translation by Kleiner (2007), p. 66 Dirichlet, Peter Gustav Lejeune (1871), Dedekind, Richard (ed.), Vorlesungen über Zahlentheorie (Lectures on Number Theory) (in German), vol. 1 (2nd ed.), Braunschweig, Germany: Friedrich Vieweg und Sohn Kleiner, Israel (2007), Kleiner, Israel (ed.), A history of abstract algebra, Birkhäuser, doi:10.1007/978-0-8176-4685-1, ISBN 978-0-8176-4684-4, MR 2347309

doi.org

Some authors also consider the fields $R$ and $C$ to be local fields. On the other hand, these two fields, also called Archimedean local fields, share little similarity with the local fields considered here, to a point that Cassels (1986, p. vi) calls them "completely anomalous". Cassels, J. W. S. (1986), Local fields, London Mathematical Society Student Texts, vol. 3, Cambridge University Press, doi:10.1017/CBO9781139171885, ISBN 0-521-30484-9, MR 0861410
Mines, Richman & Ruitenburg (1988), §II.2. See also Heyting field. Mines, Ray; Richman, Fred; Ruitenburg, Wim (1988), A course in constructive algebra, Universitext, Springer, doi:10.1007/978-1-4419-8640-5, ISBN 0-387-96640-4, MR 0919949
Kleiner (2007), p. 63 Kleiner, Israel (2007), Kleiner, Israel (ed.), A history of abstract algebra, Birkhäuser, doi:10.1007/978-0-8176-4685-1, ISBN 978-0-8176-4684-4, MR 2347309
Kiernan (1971), p. 50 Kiernan, B. Melvin (1971), "The development of Galois theory from Lagrange to Artin", Archive for History of Exact Sciences, 8 (1–2): 40–154, doi:10.1007/BF00327219, MR 1554154, S2CID 121442989
Bourbaki (1994), pp. 75–76 Bourbaki, Nicolas (1994), Elements of the history of mathematics, Springer, doi:10.1007/978-3-642-61693-8, ISBN 3-540-19376-6, MR 1290116
Dirichlet (1871), p. 42, translation by Kleiner (2007), p. 66 Dirichlet, Peter Gustav Lejeune (1871), Dedekind, Richard (ed.), Vorlesungen über Zahlentheorie (Lectures on Number Theory) (in German), vol. 1 (2nd ed.), Braunschweig, Germany: Friedrich Vieweg und Sohn Kleiner, Israel (2007), Kleiner, Israel (ed.), A history of abstract algebra, Birkhäuser, doi:10.1007/978-0-8176-4685-1, ISBN 978-0-8176-4684-4, MR 2347309
Bourbaki (1994), p. 81 Bourbaki, Nicolas (1994), Elements of the history of mathematics, Springer, doi:10.1007/978-3-642-61693-8, ISBN 3-540-19376-6, MR 1290116
Bourbaki (1994), p. 92 Bourbaki, Nicolas (1994), Elements of the history of mathematics, Springer, doi:10.1007/978-3-642-61693-8, ISBN 3-540-19376-6, MR 1290116
Lang (2002), §II.1 Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (3rd ed.), Springer, doi:10.1007/978-1-4613-0041-0, ISBN 0-387-95385-X
Eisenbud (1995), p. 60 Eisenbud, David (1995), Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, New York: Springer-Verlag, doi:10.1007/978-1-4612-5350-1, ISBN 0-387-94268-8, MR 1322960
Banaschewski (1992). Mathoverflow post Banaschewski, Bernhard (1992), "Algebraic closure without choice.", Z. Math. Logik Grundlagen Math., 38 (4): 383–385, doi:10.1002/malq.19920380136, Zbl 0739.03027
Ribenboim (1999), p. 186, §7.1 Ribenboim, Paulo (1999), The theory of classical valuations, Springer Monographs in Mathematics, Springer, doi:10.1007/978-1-4612-0551-7, ISBN 0-387-98525-5, MR 1677964
Prestel (1984), Proposition 1.22 Prestel, Alexander (1984), Lectures on formally real fields, Lecture Notes in Mathematics, vol. 1093, Springer, doi:10.1007/BFb0101548, ISBN 3-540-13885-4, MR 0769847
Prestel (1984), Theorem 1.23 Prestel, Alexander (1984), Lectures on formally real fields, Lecture Notes in Mathematics, vol. 1093, Springer, doi:10.1007/BFb0101548, ISBN 3-540-13885-4, MR 0769847
Lang (2002), Theorem V.4.6 Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (3rd ed.), Springer, doi:10.1007/978-1-4613-0041-0, ISBN 0-387-95385-X
Lang (2002), §VI.1 Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (3rd ed.), Springer, doi:10.1007/978-1-4613-0041-0, ISBN 0-387-95385-X
Lang (2002), Example VI.2.6 Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (3rd ed.), Springer, doi:10.1007/978-1-4613-0041-0, ISBN 0-387-95385-X
Jannsen & Wingberg (1982) 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, doi:10.1007/bf01393199, MR 0679774, S2CID 119378923
Eisenbud (1995), §13, Theorem A Eisenbud, David (1995), Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, New York: Springer-Verlag, doi:10.1007/978-1-4612-5350-1, ISBN 0-387-94268-8, MR 1322960
Washington (1997) Washington, Lawrence C. (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83 (2nd ed.), Springer-Verlag, doi:10.1007/978-1-4612-1934-7, ISBN 0-387-94762-0, MR 1421575
Baez (2002) Baez, John C. (2002), "The octonions", Bulletin of the American Mathematical Society, 39 (2): 145–205, arXiv:math/0105155, doi:10.1090/S0273-0979-01-00934-X, S2CID 586512

harvard.edu

ui.adsabs.harvard.edu

Jannsen & Wingberg (1982) 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, doi:10.1007/bf01393199, MR 0679774, S2CID 119378923

icm2014.org

Scholze (2014) Scholze, Peter (2014), "Perfectoid spaces and their Applications" (PDF), Proceedings of the International Congress of Mathematicians 2014, Kyung Moon SA, ISBN 978-89-6105-804-9

loc.gov

lccn.loc.gov

McCoy (1968), p. 16 McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68015225

mathoverflow.net

Banaschewski (1992). Mathoverflow post Banaschewski, Bernhard (1992), "Algebraic closure without choice.", Z. Math. Logik Grundlagen Math., 38 (4): 383–385, doi:10.1002/malq.19920380136, Zbl 0739.03027

ncsu.edu

www4.ncsu.edu

van der Put & Singer (2003), §1 van der Put, M.; Singer, M. F. (2003), Galois Theory of Linear Differential Equations, Grundlehren der mathematischen Wissenschaften, vol. 328, Springer

psu.edu

citeseerx.ist.psu.edu

Marker, Messmer & Pillay (2006), Corollary 1.2 Marker, David; Messmer, Margit; Pillay, Anand (2006), Model theory of fields, Lecture Notes in Logic, vol. 5 (2nd ed.), Association for Symbolic Logic, CiteSeerX 10.1.1.36.8448, ISBN 978-1-56881-282-3, MR 2215060

semanticscholar.org

api.semanticscholar.org

Kiernan (1971), p. 50 Kiernan, B. Melvin (1971), "The development of Galois theory from Lagrange to Artin", Archive for History of Exact Sciences, 8 (1–2): 40–154, doi:10.1007/BF00327219, MR 1554154, S2CID 121442989
Jannsen & Wingberg (1982) 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, doi:10.1007/bf01393199, MR 0679774, S2CID 119378923
Baez (2002) Baez, John C. (2002), "The octonions", Bulletin of the American Mathematical Society, 39 (2): 145–205, arXiv:math/0105155, doi:10.1090/S0273-0979-01-00934-X, S2CID 586512

tripod.com

jeff560.tripod.com

"Earliest Known Uses of Some of the Words of Mathematics (F)".

uni-bonn.de

math.uni-bonn.de

Scholze (2014) Scholze, Peter (2014), "Perfectoid spaces and their Applications" (PDF), Proceedings of the International Congress of Mathematicians 2014, Kyung Moon SA, ISBN 978-89-6105-804-9

uni-goettingen.de

resolver.sub.uni-goettingen.de

Corry (2004), p. 33. See also Fricke & Weber (1924). Corry, Leo (2004), Modern algebra and the rise of mathematical structures (2nd ed.), Birkhäuser, ISBN 3-7643-7002-5, Zbl 1044.01008 Fricke, Robert; Weber, Heinrich Martin (1924), Lehrbuch der Algebra (in German), Vieweg, JFM 50.0042.03

uni-regensburg.de

epub.uni-regensburg.de

Jannsen & Wingberg (1982) 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, doi:10.1007/bf01393199, MR 0679774, S2CID 119378923

zbmath.org

Lidl & Niederreiter (2008), Example 1.62 Lidl, Rudolf; Niederreiter, Harald (2008), Finite fields (2nd ed.), Cambridge University Press, ISBN 978-0-521-06567-2, Zbl 1139.11053
Lidl & Niederreiter (2008), Lemma 2.1, Theorem 2.2 Lidl, Rudolf; Niederreiter, Harald (2008), Finite fields (2nd ed.), Cambridge University Press, ISBN 978-0-521-06567-2, Zbl 1139.11053
Lidl & Niederreiter (2008), Theorem 1.2.5 Lidl, Rudolf; Niederreiter, Harald (2008), Finite fields (2nd ed.), Cambridge University Press, ISBN 978-0-521-06567-2, Zbl 1139.11053
Corry (2004), p. 24 Corry, Leo (2004), Modern algebra and the rise of mathematical structures (2nd ed.), Birkhäuser, ISBN 3-7643-7002-5, Zbl 1044.01008
Corry (2004), p. 33. See also Fricke & Weber (1924). Corry, Leo (2004), Modern algebra and the rise of mathematical structures (2nd ed.), Birkhäuser, ISBN 3-7643-7002-5, Zbl 1044.01008 Fricke, Robert; Weber, Heinrich Martin (1924), Lehrbuch der Algebra (in German), Vieweg, JFM 50.0042.03
Banaschewski (1992). Mathoverflow post Banaschewski, Bernhard (1992), "Algebraic closure without choice.", Z. Math. Logik Grundlagen Math., 38 (4): 383–385, doi:10.1002/malq.19920380136, Zbl 0739.03027
Warner (1989), Chapter 14 Warner, Seth (1989), Topological fields, North-Holland, ISBN 0-444-87429-1, Zbl 0683.12014
Borceux & Janelidze (2001). See also Étale fundamental group. Borceux, Francis; Janelidze, George (2001), Galois theories, Cambridge University Press, ISBN 0-521-80309-8, Zbl 0978.12004
Serre (2002) 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
Serre (1996), Chapter IV Serre, Jean-Pierre (1996) [1978], A course in arithmetic. Translation of Cours d'arithmetique, Graduate Text in Mathematics, vol. 7 (2nd ed.), Springer, ISBN 9780387900407, Zbl 0432.10001
Serre (1992) Serre, Jean-Pierre (1992), Topics in Galois theory, Jones and Bartlett Publishers, ISBN 0-86720-210-6, Zbl 0746.12001