Discriminant of an algebraic number field (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Discriminant of an algebraic number field" in English language version.

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Cohen, Diaz y Diaz & Olivier 2002 Cohen, Henri; Diaz y Diaz, Francisco; Olivier, Michel (2002), "A Survey of Discriminant Counting", in Fieker, Claus; Kohel, David R. (eds.), Algorithmic Number Theory, Proceedings, 5th International Syposium, ANTS-V, University of Sydney, July 2002, Lecture Notes in Computer Science, vol. 2369, Berlin: Springer-Verlag, pp. 80–94, doi:10.1007/3-540-45455-1_7, ISBN 978-3-540-43863-2, ISSN 0302-9743, MR 2041075
Definition 5.1.2 of Cohen 1993 Cohen, Henri (1993), A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, vol. 138, Berlin, New York: Springer-Verlag, ISBN 978-3-540-55640-4, MR 1228206
Proposition 2.7 of Washington 1997 Washington, Lawrence (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83 (2 nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94762-4, MR 1421575, Zbl 0966.11047
Narkiewicz 2004, p. 64 Narkiewicz, Władysław (2004), Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics (3 ed.), Berlin: Springer-Verlag, ISBN 978-3-540-21902-6, MR 2078267
Cohen 1993, Theorem 6.4.6 Cohen, Henri (1993), A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, vol. 138, Berlin, New York: Springer-Verlag, ISBN 978-3-540-55640-4, MR 1228206
Lemma 2.2 of Washington 1997 Washington, Lawrence (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83 (2 nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94762-4, MR 1421575, Zbl 0966.11047
Corollary III.2.12 of Neukirch 1999 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Exercise I.2.7 of Neukirch 1999 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Proposition III.2.14 of Neukirch 1999 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Theorem III.2.17 of Neukirch 1999 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Theorem III.2.16 of Neukirch 1999 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Bourbaki 1994 Bourbaki, Nicolas (1994). Elements of the history of mathematics. Translated by Meldrum, John. Berlin: Springer-Verlag. ISBN 978-3-540-64767-6. MR 1290116.
Brill 1877. Brill, Alexander von (1877), "Ueber die Discriminante", Mathematische Annalen, 12 (1): 87–89, doi:10.1007/BF01442468, JFM 09.0059.02, MR 1509928, S2CID 120947279, retrieved 2009-08-22
All facts in this paragraph can be found in Narkiewicz 2004, pp. 59, 81 Narkiewicz, Władysław (2004), Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics (3 ed.), Berlin: Springer-Verlag, ISBN 978-3-540-21902-6, MR 2078267
Neukirch 1999, §III.2 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Corollary III.2.10 of Neukirch 1999 or Proposition III.2.15 of Fröhlich & Taylor 1993 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021. Fröhlich, Albrecht; Taylor, Martin (1993), Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, ISBN 978-0-521-43834-6, MR 1215934
Voight 2008 Voight, John (2008), "Enumeration of totally real number fields of bounded root discriminant", in van der Poorten, Alfred J.; Stein, Andreas (eds.), Algorithmic number theory. Proceedings, 8th International Symposium, ANTS-VIII, Banff, Canada, May 2008, Lecture Notes in Computer Science, vol. 5011, Berlin: Springer-Verlag, pp. 268–281, arXiv:0802.0194, doi:10.1007/978-3-540-79456-1_18, ISBN 978-3-540-79455-4, MR 2467853, S2CID 30036220, Zbl 1205.11125
Section 4.4 of Serre 1967 Serre, Jean-Pierre (1967), "Local class field theory", in Cassels, J. W. S.; Fröhlich, Albrecht (eds.), Algebraic Number Theory, Proceedings of an instructional conference at the University of Sussex, Brighton, 1965, London: Academic Press, ISBN 0-12-163251-2, MR 0220701

arxiv.org

Voight 2008 Voight, John (2008), "Enumeration of totally real number fields of bounded root discriminant", in van der Poorten, Alfred J.; Stein, Andreas (eds.), Algorithmic number theory. Proceedings, 8th International Symposium, ANTS-VIII, Banff, Canada, May 2008, Lecture Notes in Computer Science, vol. 5011, Berlin: Springer-Verlag, pp. 268–281, arXiv:0802.0194, doi:10.1007/978-3-540-79456-1_18, ISBN 978-3-540-79455-4, MR 2467853, S2CID 30036220, Zbl 1205.11125

doi.org

Cohen, Diaz y Diaz & Olivier 2002 Cohen, Henri; Diaz y Diaz, Francisco; Olivier, Michel (2002), "A Survey of Discriminant Counting", in Fieker, Claus; Kohel, David R. (eds.), Algorithmic Number Theory, Proceedings, 5th International Syposium, ANTS-V, University of Sydney, July 2002, Lecture Notes in Computer Science, vol. 2369, Berlin: Springer-Verlag, pp. 80–94, doi:10.1007/3-540-45455-1_7, ISBN 978-3-540-43863-2, ISSN 0302-9743, MR 2041075
Hermite 1857. Hermite, Charles (1857), "Extrait d'une lettre de M. C. Hermite à M. Borchardt sur le nombre limité d'irrationalités auxquelles se réduisent les racines des équations à coefficients entiers complexes d'un degré et d'un discriminant donnés", Crelle's Journal, 1857 (53): 182–192, doi:10.1515/crll.1857.53.182, S2CID 120694650, retrieved 2009-08-20
Brill 1877. Brill, Alexander von (1877), "Ueber die Discriminante", Mathematische Annalen, 12 (1): 87–89, doi:10.1007/BF01442468, JFM 09.0059.02, MR 1509928, S2CID 120947279, retrieved 2009-08-22
Minkowski 1891a. Minkowski, Hermann (1891a), "Ueber die positiven quadratischen Formen und über kettenbruchähnliche Algorithmen", Crelle's Journal, 1891 (107): 278–297, doi:10.1515/crll.1891.107.278, JFM 23.0212.01, retrieved 2009-08-20
Hajir, Farshid; Maire, Christian (2002). "Tamely ramified towers and discriminant bounds for number fields. II". J. Symbolic Comput. 33: 415–423. doi:10.1023/A:1017537415688.
Voight 2008 Voight, John (2008), "Enumeration of totally real number fields of bounded root discriminant", in van der Poorten, Alfred J.; Stein, Andreas (eds.), Algorithmic number theory. Proceedings, 8th International Symposium, ANTS-VIII, Banff, Canada, May 2008, Lecture Notes in Computer Science, vol. 5011, Berlin: Springer-Verlag, pp. 268–281, arXiv:0802.0194, doi:10.1007/978-3-540-79456-1_18, ISBN 978-3-540-79455-4, MR 2467853, S2CID 30036220, Zbl 1205.11125
Martinet, Jacques (1978). "Tours de corps de classes et estimations de discriminants". Inventiones Mathematicae (in French). 44: 65–73. Bibcode:1978InMat..44...65M. doi:10.1007/bf01389902. S2CID 122278145. Zbl 0369.12007.

harvard.edu

ui.adsabs.harvard.edu

Martinet, Jacques (1978). "Tours de corps de classes et estimations de discriminants". Inventiones Mathematicae (in French). 44: 65–73. Bibcode:1978InMat..44...65M. doi:10.1007/bf01389902. S2CID 122278145. Zbl 0369.12007.

semanticscholar.org

api.semanticscholar.org

Hermite 1857. Hermite, Charles (1857), "Extrait d'une lettre de M. C. Hermite à M. Borchardt sur le nombre limité d'irrationalités auxquelles se réduisent les racines des équations à coefficients entiers complexes d'un degré et d'un discriminant donnés", Crelle's Journal, 1857 (53): 182–192, doi:10.1515/crll.1857.53.182, S2CID 120694650, retrieved 2009-08-20
Brill 1877. Brill, Alexander von (1877), "Ueber die Discriminante", Mathematische Annalen, 12 (1): 87–89, doi:10.1007/BF01442468, JFM 09.0059.02, MR 1509928, S2CID 120947279, retrieved 2009-08-22
Voight 2008 Voight, John (2008), "Enumeration of totally real number fields of bounded root discriminant", in van der Poorten, Alfred J.; Stein, Andreas (eds.), Algorithmic number theory. Proceedings, 8th International Symposium, ANTS-VIII, Banff, Canada, May 2008, Lecture Notes in Computer Science, vol. 5011, Berlin: Springer-Verlag, pp. 268–281, arXiv:0802.0194, doi:10.1007/978-3-540-79456-1_18, ISBN 978-3-540-79455-4, MR 2467853, S2CID 30036220, Zbl 1205.11125
Martinet, Jacques (1978). "Tours de corps de classes et estimations de discriminants". Inventiones Mathematicae (in French). 44: 65–73. Bibcode:1978InMat..44...65M. doi:10.1007/bf01389902. S2CID 122278145. Zbl 0369.12007.

uconn.edu

kconrad.math.uconn.edu

Conrad, Keith. "Discriminants and ramified primes" (PDF). Theorem 1.3 (Dedekind). For a number field K, a prime p ramifies in K if and only if p divides the integer discZ(OK)

uni-goettingen.de

gdz.sub.uni-goettingen.de

Dedekind 1878, pp. 30–31 Dedekind, Richard (1878), "Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Congruenzen", Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 23 (1), retrieved 2009-08-20
Dedekind's supplement X of the second edition of Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie (Dedekind 1871) Dedekind, Richard (1871), Vorlesungen über Zahlentheorie von P.G. Lejeune Dirichlet (2 ed.), Vieweg, retrieved 2009-08-05
Hermite 1857. Hermite, Charles (1857), "Extrait d'une lettre de M. C. Hermite à M. Borchardt sur le nombre limité d'irrationalités auxquelles se réduisent les racines des équations à coefficients entiers complexes d'un degré et d'un discriminant donnés", Crelle's Journal, 1857 (53): 182–192, doi:10.1515/crll.1857.53.182, S2CID 120694650, retrieved 2009-08-20
Brill 1877. Brill, Alexander von (1877), "Ueber die Discriminante", Mathematische Annalen, 12 (1): 87–89, doi:10.1007/BF01442468, JFM 09.0059.02, MR 1509928, S2CID 120947279, retrieved 2009-08-22
Kronecker 1882. Kronecker, Leopold (1882), "Grundzüge einer arithmetischen Theorie der algebraischen Grössen", Crelle's Journal, 92: 1–122, JFM 14.0038.02, retrieved 2009-08-20
Minkowski 1891a. Minkowski, Hermann (1891a), "Ueber die positiven quadratischen Formen und über kettenbruchähnliche Algorithmen", Crelle's Journal, 1891 (107): 278–297, doi:10.1515/crll.1891.107.278, JFM 23.0212.01, retrieved 2009-08-20

worldcat.org

search.worldcat.org

Cohen, Diaz y Diaz & Olivier 2002 Cohen, Henri; Diaz y Diaz, Francisco; Olivier, Michel (2002), "A Survey of Discriminant Counting", in Fieker, Claus; Kohel, David R. (eds.), Algorithmic Number Theory, Proceedings, 5th International Syposium, ANTS-V, University of Sydney, July 2002, Lecture Notes in Computer Science, vol. 2369, Berlin: Springer-Verlag, pp. 80–94, doi:10.1007/3-540-45455-1_7, ISBN 978-3-540-43863-2, ISSN 0302-9743, MR 2041075
Manin, Yu. I.; Panchishkin, A. A. (2007), Introduction to Modern Number Theory, Encyclopaedia of Mathematical Sciences, vol. 49 (Second ed.), p. 130, ISBN 978-3-540-20364-3, ISSN 0938-0396, Zbl 1079.11002

zbmath.org

Manin, Yu. I.; Panchishkin, A. A. (2007), Introduction to Modern Number Theory, Encyclopaedia of Mathematical Sciences, vol. 49 (Second ed.), p. 130, ISBN 978-3-540-20364-3, ISSN 0938-0396, Zbl 1079.11002
Proposition 2.7 of Washington 1997 Washington, Lawrence (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83 (2 nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94762-4, MR 1421575, Zbl 0966.11047
Koch 1997, p. 11 Koch, Helmut (1997), Algebraic Number Theory, Encycl. Math. Sci., vol. 62 (2nd printing of 1st ed.), Springer-Verlag, ISBN 3-540-63003-1, Zbl 0819.11044
Lemma 2.2 of Washington 1997 Washington, Lawrence (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83 (2 nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94762-4, MR 1421575, Zbl 0966.11047
Corollary III.2.12 of Neukirch 1999 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Exercise I.2.7 of Neukirch 1999 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Proposition III.2.14 of Neukirch 1999 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Theorem III.2.17 of Neukirch 1999 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Theorem III.2.16 of Neukirch 1999 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Brill 1877. Brill, Alexander von (1877), "Ueber die Discriminante", Mathematische Annalen, 12 (1): 87–89, doi:10.1007/BF01442468, JFM 09.0059.02, MR 1509928, S2CID 120947279, retrieved 2009-08-22
Kronecker 1882. Kronecker, Leopold (1882), "Grundzüge einer arithmetischen Theorie der algebraischen Grössen", Crelle's Journal, 92: 1–122, JFM 14.0038.02, retrieved 2009-08-20
Minkowski 1891a. Minkowski, Hermann (1891a), "Ueber die positiven quadratischen Formen und über kettenbruchähnliche Algorithmen", Crelle's Journal, 1891 (107): 278–297, doi:10.1515/crll.1891.107.278, JFM 23.0212.01, retrieved 2009-08-20
Minkowski 1891b. Minkowski, Hermann (1891b), "Théorèmes d'arithmétiques", Comptes rendus de l'Académie des sciences, 112: 209–212, JFM 23.0214.01
Stickelberger 1897. Stickelberger, Ludwig (1897), "Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper", Proceedings of the First International Congress of Mathematicians, Zürich, pp. 182–193, JFM 29.0172.03
Neukirch 1999, §III.2 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Corollary III.2.10 of Neukirch 1999 or Proposition III.2.15 of Fröhlich & Taylor 1993 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021. Fröhlich, Albrecht; Taylor, Martin (1993), Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, ISBN 978-0-521-43834-6, MR 1215934
Koch 1997, pp. 181–182 Koch, Helmut (1997), Algebraic Number Theory, Encycl. Math. Sci., vol. 62 (2nd printing of 1st ed.), Springer-Verlag, ISBN 3-540-63003-1, Zbl 0819.11044
Voight 2008 Voight, John (2008), "Enumeration of totally real number fields of bounded root discriminant", in van der Poorten, Alfred J.; Stein, Andreas (eds.), Algorithmic number theory. Proceedings, 8th International Symposium, ANTS-VIII, Banff, Canada, May 2008, Lecture Notes in Computer Science, vol. 5011, Berlin: Springer-Verlag, pp. 268–281, arXiv:0802.0194, doi:10.1007/978-3-540-79456-1_18, ISBN 978-3-540-79455-4, MR 2467853, S2CID 30036220, Zbl 1205.11125
Martinet, Jacques (1978). "Tours de corps de classes et estimations de discriminants". Inventiones Mathematicae (in French). 44: 65–73. Bibcode:1978InMat..44...65M. doi:10.1007/bf01389902. S2CID 122278145. Zbl 0369.12007.