Class Groups |
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| title | journal | file |
| Kuroda's Class Number Formula | Acta Arith. 66 (1994), 245-260 | |
| Class groups of cyclotomic number fields I | Acta Arith. 72 (1995), 347-359 | arxiv |
| Class groups of cyclotomic number fields II | Acta Arith. 84 (1998), 59-70 | arxiv |
| Class groups of cyclotomic number fields III | Acta Arith. 131 (2008), 255-266 | arxiv |
| The 4-class group of real quadratic number fields | unpublished | arxiv |
| On class number 1 problems of some normal CM fields of degree 24 | J. Theor. Nombres Bordx. 11 (1999), 387-406 | |
| Galois Action on Class Groups | J. Algebra 264 (2003), 553-564 | arxiv |
| Class Groups of Dihedral Extensions
Daniel Mayer has pointed out that there is something wrong with Prop. 8.6.b) (which is Thm. 6 in the arxiv-version). | Math. Nachr 278 (2005), 679-691 | arxiv |
| Why is the class number of Q(11^(1/3)) even?
Jianing Li has pointed out to me that m must not be a square in Lemma 1.2.; otherwise m=4 is a counterexample. | Bohemian J. Math. 138 (2013), No. 2, 149-163 | arxiv |
| Binomial Squares in Pure Cubic Number Fields | J. Theor. Nombres Bordeaux 24 (2012), 691-704 | arxiv |
| Relations in the 2-Class Group of Quadratic Number Fields | J. Austral. Math. Soc. 93 (2012), 115-120 | arxiv |
| Simerka - Quadratic Forms and Factorization | LMS J. Comput. Math. 16 (2013), 118--129 | arxiv |
| The Ambiguous Class Number Formula Revisited | J. of the Ramanujan Math. Soc. 415-421 | arxiv |
Reciprocity Laws |
| Rational Quartic Reciprocity | Acta Arith. 67 (1994), 387--390 | arxiv |
| Rational Quartic Reciprocity II | Acta Arith. 80 (1997), 273--276 | arxiv |
| Reciprocity Laws. From Euler to Eisenstein | Springer Verlag | html |
| A Supplement to Scholz's reciprocity law | Acta Arith. 129 (2007), 319--324 | arxiv |
| Cubic residuacity and quadratic forms | Lecture Braunschweig 2016 | |
| Harbingers of Artin's reciprocity law I. The Continuing Story of Auxiliary Primes | unpublished | arxiv |
| Harbingers of Artin's reciprocity law II. Irreducibility of Cyclotomic Polynomials | unpublished | arxiv |
| Harbingers of Artin's reciprocity law III. Gauss's Lemma and Artin's Transfer | unpublished | arxiv |
| Harbingers of Artin's reciprocity law IV. Bernstein's Reciprocity Law | unpublished | arxiv |
| Hermite's identity and the quadratic reciprocity law | Elemente der Mathematik | |
| Bibliography on Reciprocity Laws | www | html |
| A list of proofs of the Quadratic Reciprocity Law. | www | html |
Euclidean Number Fields |
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| The Euclidean Algorithm in Algebraic Number Fields; | Expo. Math. 13 (1995), 385-416. | |
| The Euclidean Algorithm in cubic number fields (with Stefania Cavallar) | Proceedings Number Theory Eger 1996, (Györy, Pethö, Sos eds.), Gruyter 1998, 123-146 | arxiv |
| On Gauss Bounds of quadratic extensions of imaginary quadratic Euclidean number fields | Publ. Math. Debrecen 50 (1997), 365-368 | |
| Euclidean Windows (with Stefania Cavallar) | LMS J. Comput. Math. 3 (2000), 335-355 | arxiv |
| Quartic Euclidean Number Fields | unpublished | arxiv |
| An Application of the Dedekind-Hasse Criterion | unpublished | arxiv |
| Online Bibliography on Euclidean Rings | www | html |
Class Field Theory |
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| On 2-class field towers of imaginary quadratic number fields; | J. Theor. Nombres Bordeaux 6 (1994), 261-272 | arxiv |
| Construction of Hilbert 2-Class Fields | Short introduction to my thesis | |
| On unramified quaternion extensions of quadratic number fields | J. Theor. Nombres Bordeaux 9 (1997), 51-68 | arxiv |
| Imaginary quadratic number fields with cyclic Cl2(k1) (with Elliot Benjamin and C. Snyder) | J. Number Theory 67 (1997), 229-245 | |
| Real quadratic number fields with Abelian Gal(k2/k) (with Elliot Benjamin and Chip Snyder) | J. Number Theory 73 (1998), 182-194 | |
| The class number one problem for some non-abelian normal CM-fields of degree 24 (with S. Louboutin and R. Okazaki) | J. Théor. Nombres Bordx. 11, No. 2, 387-406 (1999) | |
| On the 2-class field tower of some imaginary quadratic number fields | Abh. Math. Sem. Hamburg 67 (1997), 205-214 | arxiv |
| Imaginary quadratic fields k with Cl2(k) = (2,2m) and d(Cl2(k1)) = 2 (with Elliot Benjamin and Chip Snyder) | Pac. J. Math. 198 (2001), 15-31 | |
| Imaginary quadratic fields with Cl2(k) = (2,2,2) (with Elliot Benjamin and C. Snyder) | J. Number Theory 103 (2003), 38-70 | |
| A survey on Class Field Towers | www | |
| Postscript to the English translation of Koch's Galois theory of p-extensions (with H. Koch) | Springer-Verlag, August 2002 | |
| Selmer groups and quadratic reciprocity Alvaro Lozano-Robledo pointed out to me that the last sentence in Prop. 4.5 is nonsense. | Abh. Math. Sem. Hamburg 2007 | arxiv |
| Kronecker-Weber via Stickelberger | J. Théor. Nombres Bordeaux 17 (2005), 579--582 | arxiv |
| On the unit group of some multiquadratic number fields (with E. Benjamin, C. Snyder) | Pac. J. Math. 230 (2007), 27-40 | |
| Grün's theorems and class groups | Arch. Math. 96 (2011), 135-138 | arxiv |
| Hilbert 2-Class Fields and 2-Descent | preprint | arxiv |
| On the Normality of Hilbert Class Fields | preprint | |
Elliptic Curves |
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| On Tate-Shafarevich groups of some elliptic curves; | Algebraic number theory and Diophantine analysis (Graz, 1998), 277-291 (2000) | arxiv |
| A note on Pépin's counter examples to the Hasse principle for curves of genus 1 | Abh. Math. Sem. Hamburg 69 (1999), 335-345 | arxiv |
| Some families of noncongruent numbers | Acta Arithmetica 110 (2003), 15-36 | arxiv |
| On Tate-Shafarevich groups of y2 = x(x2 - k2) (with R. Mollin) | Acta Math. Univ. Comenianae 72 (2003), 73-80 | |
| Binary Quadratic Forms and Counterexamples to Hasse's Local-Global Principles | submitted | arxiv |
Historical |
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| Introduction (with Norbert Schappacher) to the English translation of Hilbert's Zahlbericht by I. Adamson | Springer-Verlag | |
| Development of the principal genus theorem (from Gauss to Noether) | Springer-Verlag 2003, proceedings of a conference on 200 years of Disquisitiones Arithmeticae in Oberwolfach 2001 | |
| Helmut Hasse und Emmy Noether. Die Korrespondenz 1925--1935 (with P. Roquette) | Universitätsverlag Göttingen 2006 | html |
| Jacobis Vorlesungen über Zahlentheorie aus dem Wintersemester 1836/37 (with H. Pieper) | Algorismus 62, Rauner Verlag Augsburg 2007 | html |
| Emil Artin und Helmut Hasse. Die Korrespondenz 1923 -- 1934. G. Frei, P. Roquette, unter Mitwirkung von F. Lemmermeyer | Universitätsverlag Göttingen 2006 | html |
| Zur Zahlentheorie der Griechen I: Euklids Fundamentalsatz der Arithmetik | Math. Semesterberichte 55 (2008), 181-195 | |
| Zur Zahlentheorie der Griechen II: Gauss'sche Lemmas und Rieszsche Ringe | Math. Semesterberichte 56 (2009), 39-51 | |
| Jacobi and Kummer's ideal numbers | Abh. Math. Sem. Hamburg 79 (2009), 165-187 | arxiv |
| Euler, Goldbach, and "Fermat's Theorem" | Elem. Math. 2010 | arxiv |
| Die mathematischen Tagebücher von Hasse 1923-1935 | Universitätsverlag Göttingen 2013 | html |
| Leonardo da Vinci's Proof of the Theorem of Pythagoras | The College Mathematics Journal 47, No. 5 (November 2016), pp. 361-364 | |
| Sumerian traces in Archmimedes' Cattle Problem | ||
| Emil Artin and Helmut Hasse. The Correspondence 1923 -- 1934. G. Frei, F. Lemmermeyer, P. Roquette, | Springer 2014 | html pdf |
| The Correspondence Euler - Goldbach (with M. Mattmüller) | Euler's Opera Omnia, Series IV-A.4 (2014) | html pdf |
| Dedekinds Arbeit über rein kubische Zahlkörper | Proceedings Dedekind Conference Braunschweig 2016 | |
| Hilberts Zahlbericht | Jahresbericht DMV (2017) | |
| Class numbers of abelian number fields | Introduction to the english translation of Hasse's book by Mikihito Hirabayashi (2017) | |
| HIstory of Number Theory | Collection of online material on the history of number theory, including webliographies and letters between Hasse, R. Brauer and E. Noether. | html |
Conics and Affine Algebraic Varieties |
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| Kreise und Quadrate modulo p | Mathem. Semesterberichte 47 (2000), 51-73 | |
| Conics: a Poor Man's Elliptic Curves | preprint | arxiv |
| Higher Descent on Pell Conics I. From Legendre to Selmer | preprint | arxiv |
| Higher Descent on Pell Conics II. Two Centuries of Missed Opportunities | preprint | arxiv |
| Higher Descent on Pell Conics III. The First $2$-Descent | preprint | arxiv |
| Counterexamples to the Hasse principle: An Elementary Introduction (with W. Aitken) | Amer. Math. Monthly 118 (2011), 610-628 | arxiv |
| Arithmetic of Pell Surfaces (with S. Hambleton) | Acta Arithmetica 146 (2011), 1-12 | arxiv |
| Modular Pell conics | preprint | |
| Parametrization of Algebraic Curves from a number theorist's point of view. | Amer. Math. Mon. 119 (2012), 573-583 | arxiv |
| The Pell Equation | webliography up to 1945 | html |
Schulmathematik |
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| Kritik an der Schulmathematik | Eigene Seite | Zur Zahlengeometrie der Babylonier | Mathematik-Information 2018 | Polynome: Grenzwertfreie Analysis | Mathematik-Information 2020 |
The Rest |
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| Small norms in a quadratic field; an introductory section to the paper on simplest cubic fields that was omitted in the published version | unpublished | arxiv |
| On Simplest Cubic Fields (with A. Pethö); | Manuscripta Math. 88 (1995), 53--58 | arxiv |
| The Snake Lemma (new version Nov. 2002) | unpublished | |
| Infinitude of primes | ||
| Unsolved problems in Algebraic Number Theory | unpublished | |
| Composite values of irreducible polynomials | Elemente der Mathematik, 2018 | |
| Hermite's identity | unpublished | |
| Hurwitz und die Eulersche Identität | ||
Translations of some articles |
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| Boas Erez: an introduction to the fascinating topic of normal integral bases (in German; hey, the original is Italian!) | Representations of groups in algebraic number theory. An introduction. (Italian) Mathematics, Proc. Colloq., Locarno/Italy 1988-1989, Note Mat. Fis. 3, vol. 4, 41-65 (1990). | |
| Coates & and Poitou: An account of the history of the main conjecture as proved by Mazur and Wiles | Du nouveau sur les racines de l'unité (French) Gaz. Math., Soc. Math. Fr. 15, 5-26 (1980). | |
| Serre: On congruences involving Ramanujan's tau-function and their interpretation in terms of l-adic representations | Une interpretation des congruences relatives a la fonction de Ramanujan (French) Semin. Delange-Pisot-Poitou 9 (1967/68), Theorie Nombres, No.14, 17p. (1969) | |
| Velu: Exposition of Serre's article on the Galois properties of division points on elliptic curves | Groupes de Galois attachés aux points d'ordre fini des courbes elliptiques sur un corps de nombres (d'après J. P. Serre). (French) Sem. Theorie Nombres 1971-1972, Univ. Bordeaux, No.30, 12 p. (1972) | |
| M. Waldschmidt: An introduction to complex multiplication of elliptic curves. | notes rédigées par A. Faisant, R. Lardon et G. Philibert, Sém Arithm. Univ. St Etienne, 1981-82, No 4, 23 pp. | |
| Guy Henniart: The congruent number problem and its connections with elliptic curves and modular forms. | some journal of the SMF | |
| Koch: Über den 2-Klassenkörperturm eines quadratischen Zahlkörpers | J. Reine Angew. Math. 214/215 (1964), 201--206. | |
| Hasse: Vorlesungen über Klassenkörpertheorie. Teil I. Galois Theory | Physica Verlag Würzburg | |
| Jacques Herbrand: Determination of the ramification groups of a field from those of an extension (Détermination des groupes de ramification d'un corps à partir de ceux d'un sur-corps) | C. R. Acad. Sci. Paris 191 (1930), 980-982 | |
Last update: April 2020