Bibliography on Euclidean Rings

  1. A. G. Agargün, Nagata's theorem, Math. Japon. 43 (1996), no. 3, 421--423; MR
  2. A. G. Agargün, On Euclidean rings, Proyecciones 16 (1997), no. 1, 23--36; MR
  3. A. G. Agargün, C. R. Fletcher, Euclidean rings, Turkish. J. Math. 19 (1995), 291--299 Zbl; MR
  4. R. Akhtar, Cyclotomic Euclidean Number Fields, Senior Thesis, Harvard Univ. 1995
  5. K. Amano, A remark on Euclidean algorithm, Bull. Fac. Gen. Ed. Gifu Univ. 1981, no. 17 (1982), 55--56 MR
  6. K. Amano, Correction to: "A remark on Euclidean algorithm", Bull. Fac. Gen. Ed. Gifu Univ. No. 18 (1982), 111 MR
  7. K. Amano, A note on Euclidean ring, Bull. Fac. Gen. Ed. Gifu Univ. No. 20 (1984), 13--15 (1985); MR
  8. K. Amano, On 2-stage Euclidean ring and Laurent series, Bull. Fac. Gen. Ed. Gifu Univ. No. 22 (1986), 83--86 (1987); MR
  9. K. Amano, On the Euclidean modules, Bull. Fac. Gen. Ed. Gifu Univ. No. 27 (1991), 63--66; MR
  10. K. Amano, K. Iwata, A Euclidean algorithm on Mn(\Delta) , Bull. Fac. Gen. Ed. Gifu Univ. No. 23, (1987), 25--29 (1988); MR
  11. V. S. Atabekyan, The Euclidean algorithm for integer quaternions and the Lagrange theorem (Russ.), Erevan. Gos. Univ. Uchen. Zap. Estestv. Nauki (1993), no. 1 (179), 125--127; MR
  12. M. Baica, Baica's general Euclidean algorithm (BGEA) and the solution of Fermat's last theorem, Notes Number Theory Discrete Math. 1 (1995), no. 3, 120--134 MR
  13. R. P. Bambah, Cambridge University Thesis 1950
  14. R. P. Bambah, Nonhomogeneous binary quadratic forms I, Acta Math. 86 (1951), 1--29; Zbl; MR
  15. R. P. Bambah, Nonhomogeneous binary quadratic forms II, Acta Math. 86 (1951), 31--56; MR
  16. E.S. Barnes, Non-homogeneous binary quadratic forms, Quart. J. Math., Oxford Ser. (2) 1 (1950), 199--210; MR
  17. E.S. Barnes, Note on non-homogeneous linear forms, Proc. Cambridge Phil. Soc. 49 (1953), 360--362 MR
  18. E.S. Barnes, The inhomogeneous minima of binary quadratic forms IV, Acta Math. 92 (1954), 235--264
  19. E.S. Barnes, H.P.F. Swinnerton-Dyer, The inhomogeneous minima of binary quadratic forms I, Acta Math. 87 (1952), 259--323; Zbl
  20. E.S. Barnes, H.P.F. Swinnerton-Dyer, The inhomogeneous minima of binary quadratic forms II, Acta Math. 88 (1952), 279--316; Zbl
  21. E.S. Barnes, H.P.F. Swinnerton-Dyer, The inhomogeneous minima of binary quadratic forms III, Acta Math. 92 (1954), 199-234; Zbl
  22. E. Bedocchi, L'anneau $\Z(\sqrt{14}\,)$ et l'algorithme Euclidien, Manuscripta math. 53 (1985), 199--216; Zbl
  23. E. Bedocchi, On the second minimum of a quadratic form and its applications, (Ital.), Riv. Mat. Univ. Parma (4) 15 (1989), 175--190; Zbl
  24. H. Behrbohm, L. Redei, Der Euklidische Algorithmus in quadratischen Zahlkörpern, J. Reine Angew. Math. 174 (1936), 192--205; Zbl
  25. E. Berg, Über die Existenz eines Euklidischen Algorithmus in quadratischen Zahlkörpern, Kungl. Fysiogr. Saellsk. i Lund Förh. 5 (1935), 53--58; Zbl
  26. D. Berend, W. Moran, The inhomogeneous minimum of binary quadratic forms, Math. Proc. Camb. Phil. Soc. 112 (1992), 7--19; Zbl
  27. G. B. Birkhoff, Note on certain quadratic number systems for which factorization is unique, Amer. Math. Monthly 13 (1906), 156--159
  28. J. Blöhmer, Euklidische Ringe, Diplomarbeit TU Berlin, 1989
  29. B. Bougaut, Anneaux quasi-euclidiens, Thèse, Université de Poitiers 1976
  30. B. Bougaut, Anneaux quasi-euclidiens, C. R. Acad. Sci. Paris 284 (1977), 133--136; Zbl
  31. B. Bougaut, Algorithme explicite pour la recherche du P.G.C.D. dans certains anneaux principaux d'entiers de corps de nombres, Theoret. Comput. Sci. 11 (1980), 207--220; Zbl
  32. G. Branchini, Algoritmo del massimo comun divisore nel corpo delle radici quinte dell'unita, Atti della Reale Accademia dei Lincei, Roma (5) 32-1 (1923), 68--72
  33. A. Brauer, On the non-existence of the Euclidean algorithm in certain quadratic number fields, Amer. J. Math. 62 (1940), 697--716; Zbl
  34. A. Brudnyi, On Euclidean domains, Commun. Algebra 21 (1993), 3327-3336; Zbl
  35. V. Brun, Music and Euclidean algorithms (Norwegian), Nordisk Mat. Tidskr. 9 (1961), 29--36; MR
  36. V. Brun, Euclidean algorithms and musical theory, Enseignement Math. (2) 10 (1964), 125--137; MR
  37. H. H. Brungs, Left Euclidean rings, Pacific J. Math. 45 (1973), 27--33
  38. E. Cahen, Sur une note de M. Fontené relative aux entiers algebriques de la forme $x+y\sqrt{-5}$, Nouv. ann. math. (4) 3 (1903), 444--447
  39. O. A. Campoli, A principal ideal domain that is not a Euclidean domain, American Math. Monthly 95 (1988), 868--871; Zbl
  40. D. A. Cardon, A Euclidean ring containing $\Z[\sqrt{14}\,]$, C. R. Math. Rep. Acad. Sci. Canada 19 (1997), 28--32
  41. J. W. S. Cassels, The lattice properties of asymmetric hyperbolic regions I, Proc. Cambridge Phil. Soc. 44 (1948), 1--7
  42. J. W. S. Cassels, The lattice properties of asymmetric hyperbolic regions II, Proc. Cambridge Phil. Soc. 44 (1948), 145--154
  43. J. W. S. Cassels, The lattice properties of asymmetric hyperbolic regions III, Proc. Cambridge Phil. Soc. 44 (1948), 457--462
  44. J. W. S. Cassels, The inhomogeneous minima of binary quadratic, ternary cubic, and quaternary quartic forms, Proc. Cambridge Phil. Soc. 48 (1952), 72--86; Zbl; Addendum: ibid., 519--520; Zbl; Corr.: F.J. van der Linden 1983
  45. J. W. S. Cassels, Yet another proof of Minkowski's theorem on the product of two inhomogeneous linear forms, Proc. Cambridge Phil. Soc. 49 (1953), 365--366; Zbl
  46. I. Castro Chadid, Euclidean quadratic fields (Span.), Bol. Mat. 18 (1984), 1--3, 7--21; Zbl
  47. M. A. Cauchy, Memoire sur de nouvelles formules relatives a la théorie des polynomes radicaux, et sur le dernier théorème de Fermat I, Comptes rendus Paris 24 (1847), 469 ff; see also \OE uvres X, extr. no. 358, 240--254;
  48. M. A. Cauchy, Memoire sur de nouvelles formules relatives a la théorie des polynomes radicaux, et sur le dernier théorème de Fermat II, Comptes rendus Paris 24 (1847), 516 ff; see also \OE uvres X, extr. no. 359, 254--268;
  49. S. Cavallar, Alcuni esempi di campi di numeri cubici non euclidei rispetto alla norma ma euclidei rispetto ad una norma pesata, Tesi di Laurea, Univ. Trento, 1995
  50. S. Cavallar, F. Lemmermeyer, The Euclidean Algorithm in Cubic Number Fields, Proceedings Number Theory Eger 1996, (Györy, Pethö, Sos eds.), Gruyter 1998, 123--146; ps file; Zbl
  51. S. Cavallar, F. Lemmermeyer, Euclidean Windows, dvi file; submitted
  52. J.-P. Cerri, Letter from Nov. 21, 1997
  53. H. Chatland, On the Euclidean algorithm in quadratic number fields, Bull. Amer. Math. Soc. 55 (1949), 948--953; Zbl
  54. H. Chatland, H. Davenport, Euclids algorithm in real quadratic fields, Canad. J. Math. 2 (1950), 289--296; Zbl
  55. T. Chella, Dimostrazione dell'essisenza di un algoritmo delle divisioni successive per alcuni corpi circolari, Annali di Matematica pura ed applicata (4) {\bf} 1 (1924), 199--218
  56. W. Y. Chen, M. G. Leu, On Nagata's pairwise algorithm, J. Algebra 165 (1994), no. 1, 194--203; MR; Zbl
  57. V. Cioffari, The Euclidean condition in pure cubic and complex quartic fields, Math. Comp. 33 (1979), 389--398; Zbl; MR
  58. D. A. Clark, The Euclidean algorithm for Galois extensions of the rational numbers, Ph. D. thesis, McGill University, Montreal (1992)
  59. D. A. Clark, A quadratic field which is Euclidean but not norm-Euclidean, Manuscripta math. 83 (1994), 327--330; Zbl; MR
  60. D. A. Clark, Non-Galois cubic fields which are Euclidean but not norm-Euclidean, Math. Comp. 65 (1996), 1675--1679; Zbl; MR
  61. D. A. Clark, On k-stage Euclidean Algorithms for Galois extensions of Q, Manuscripta Math. 90 (1996), 149--153; Zbl; MR
  62. D. A. Clark, M. R. Murty, The Euclidean Algorithm in Galois Extensions of Q, J. Reine Angew. Math. 459 (1995), 151--162; Zbl; MR
  63. L. E. Clarke, On the product of three non-homogeneous linear forms, Proc. Cambridge Phil. Soc. 47 (1951), 260--265; Zbl
  64. L. E. Clarke, Non-homogeneous linear forms associated with algebraic fields, Quart. J. Math. (2) 2 (1951), 308--315
  65. H. J. Claus, Über die Partialbruchzerlegung in nicht notwendig kommutativen euklidischen Ringen, J. Reine Angew. Math. 194 (1955), 88--100
  66. H. Cohen, Hermite and Smith normal form algorithms over Dedekind domains, Math. Comp. 65 (1996), 1681--1699; Zbl
  67. H. Cohn, J. Deutsch, Use of a computer scan to prove that $\Q(\sqrt{2+\sqrt2})$ and $\Q(\sqrt{3+\sqrt2})$ are Euclidean, Math. Comp. 46 (1986), 295--299; Zbl
  68. P. M. Cohn, On a generalization of the Euclidean algorithm, Proc. Cambridge Phil. Soc. 57 (1961), 18--30
  69. P. M. Cohn, On the structure of the GL(2) of a ring, I.H.E.S. Publ. Math. 30 (1966), 365--413; Zbl
  70. P. M. Cohn, A presentation of SL(2) for Euclidean imaginary quadratic fields, Mathematika 15 (1968), 156--163
  71. P. M. Cohn, Euclid's algorithm---since Euclid, Math. Medley 19 (1991), no. 2, 65--72;
  72. A. Coja-Oghlan, Berechnung kubischer euklidischer Zahlkörper, Diplomarbeit FU Berlin, 1999
  73. A. J. Cole, J. T. Davie, A game based on the Euclidean algorithm and a winning strategy for it, Math. Gaz. 53 (1969), 354--357; MR
  74. V. Collazo, A. Correa, M. Pablo, Revisiting Euclid's algorithm: a geometric point of view, Proceedings of the Twenty-fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1994). Congr. Numer. 104 (1994), 7--17.
  75. M. J. Collison, The unique factorization theorem: from Euclid to Gauss, Math. Mag. 53 (1980), 96--100
  76. G. Cooke, The weakening of the Euclidean property for integral domains and application to algebraic number theory I, J. Reine Angew. Math. 282 (1976), 133--156; Zbl
  77. G. Cooke, The weakening of the Euclidean property for integral domains and application to algebraic number theory II, J. Reine Angew. Math. 283 (1977), 71--85; Zbl343.13008
  78. G. Cooke, P. J. Weinberger, On the construction of division chains in algebraic number fields with application to SL(2), Comm. Algebra 2 (1975), 481--524; Zbl
  79. M. Cugiani, Osservazioni relative alla questioni dell'esistenza di un algoritmo euclidei nei campi quadratici, Boll. della Unione Math. Italiana 3 (1948), 136--141
  80. M. Cugiani, I campi quadratici e l'algoritmo Euclideo, Periodico di Math. 28 (1950), 52--62, 114--129
  81. H. Davenport, On the product of three homogeneous linear forms I, J. London Math. Soc. 13 (1938), 139--145; see also Coll. Works I, 7--13
  82. H. Davenport, On the product of three homogeneous linear forms II, Proc. London Math. Soc. (2) 44 (1938), 412--431; see also Coll. Works I, 14--33
  83. H. Davenport, On the product of three homogeneous linear forms III, Proc. London Math. Soc. (2) 45 (1939), 98--125; see also Coll. Works I, 34--61
  84. H. Davenport, Note on the product of three homogeneous linear forms, J. London Math. Soc. 16 (1941), 98--101; see also Coll. Works I, 62--65
  85. H. Davenport, On the product of three homogeneous linear forms IV, Proc. Cambridge Phil. Soc. 39 (1943), 1--21; see also Coll. Works I, 66-68
  86. H. Davenport, Non-homogeneous binary quadratic forms I, Proc. Ned. Akad. Wet. 49 (1946), 815--821; Zbl; see also Coll. Works I, 167--173; Zbl
  87. H. Davenport, Non-homogeneous binary quadratic forms II, Proc. Ned. Akad. Wet. 50 (1947), 378--389; Zbl; see also Coll. Works I, 174--185
  88. H. Davenport, Non-homogeneous binary quadratic forms III, Proc. Ned. Akad. Wet. 50 (1947), 484--491; Zbl; see also Coll. Works I, 186--193
  89. H. Davenport, Non-homogeneous binary quadratic forms IV, Proc. Ned. Akad. Wet. 50 (1947), 741-749; Zbl; see also Coll. Works I, 194--211
  90. H. Davenport, On the product of three non-homogeneous linear forms, Proc. Cambridge Phil. Soc. 43 (1947), 137--152; see also Coll. Works I, 212--227; Zbl
  91. H. Davenport, Indefinite binary quadratic forms, Quart. J. Math. Oxford (2) 1 (1950), 54--62; see also Coll. Works I, 395--403
  92. H. Davenport, Euclid's algorithm in cubic fields of negative discriminant, Acta Math. 84 (1950), 159--179; see also Coll. Works I, 374--394
  93. H. Davenport, Euclid's algorithm in certain quartic fields, Trans. Amer. Math. Soc. 68 (1950), 508--532; Zbl; see also Coll. Works I, 404--428
  94. H. Davenport, L'algorithme d'Euclide dans certains corps algebriques, Coll. intern. du Centre de la recherche sci. XXIV; Zbl; Algebrè et theorie des nombres (1950), 41--43
  95. H. Davenport, Indefinite binary quadratic forms and Euclid's algorithm in real quadratic fields, Proc. London Math. Soc. (2) 53 (1951), 65--82; Zbl; see also Coll. Works I, 344-361
  96. H. Davenport, Linear forms associated with an algebraic number field, Quart. J. Math. (2) 3 (1952), 32--41; see also Coll. Works II, 552--561
  97. H. Davenport, H. P. F. Swinnerton-Dyer Products of n inhomogeneous linear forms, Proc. London Math. Soc. (3) 5 (1955), 474--499; see also Coll. Works of H. Davenport II, 566--591
  98. H. Décoste, Sur les anneaux euclidiens et principaux, Diss. Univ. Montreal 1978
  99. H. Décoste, Des anneaux principaux non euclidiens, Ann. Sci. Math. Quebec 5 (1981), 103--114
  100. R. Dedekind, Supplement to L. Dirichlet, ``Vorlesungen über Zahlentheorie'', Braunschweig 1893
  101. K. Dennis, B. Magurn, L. Vaserstein, Generalized Euclidean group rings, J. Reine Angew. Math. 351 (1984), 113--128
  102. L. E. Dickson, Algebren und ihre Zahlentheorie, Zürich-Leipzig 1927
  103. A. Dietz, Ein kubischer Zahlkörper, welcher euklidisch aber nicht norm-euklidisch ist, Staatsexamensarbeit, FU Berlin
  104. L. Dirichlet, Recherches sur les forms quadratiques à coéfficients et à indeterminées complexes, J. Reine Angew. Math. 24 (1842), 291--371; see also Werke I, 533--618
  105. L. Dirichlet, Ueber die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen, J. Reine Angew. Math. 40 (1850), 209--277; see also Werke II, 27-48
  106. D. W. Dubois, A. Steger, A note on the division algorithms in imaginary quadratic number fields, Canad. J. Math. 19 (1958), 285-286
  107. F. J. Dyson, On the product of four non-homogeneous linear forms, Ann. Math. 49 (1948), 82--109
  108. S. Egami, Euclid's algorithm in pure quartic fields, Tokyo J. Math. 2 (1979), 379--385
  109. S. Egami, On finiteness of numbers of Euclidean fields in some classes of number fields, Tokyo J. Math. 7 (1984), 183--198
  110. R. B. Eggleton, C. B. Lacampagne, J. L. Selfridge, Euclidean quadratic fields, Amer. Math. Monthly 99 (1992), 829--837
  111. G. F. Eisenstein, Über einige allgemeine Eigenschaften der Gleichung, von welcher die Teilung der ganzen Lemniskate abhängt, nebst Anwendungen derselben auf die Zahlentheorie, J. Reine Angew. Math. 39 (1850), 224--287; see also Math. Werke II, 556--619
  112. V. Ennola, On the first inhomogeneous minimum of indefinite binary quadratic forms and Euclid's algorithm in real quadratic fields, Ann. Univ. Turku. Ser. A I 28 (1958), 58pp; Zbl
  113. P. Erdös, Ch. Ko, Note on the Euclidean algorithm, J. London Math. Soc. 13 (1938), 3--8
  114. H.R.P. Ferguson, R.W. Forcade, Mulidimensional Euclidean algorithms, J. Reine Angew. Math. 334 (1982), 171--181
  115. A. K. Feyziglu, A very simple proof that $\Z[\frac12(1+\sqrt{-19})]$ is a principal ideal domain, Doga Mat. 16 (1992), 62--68
  116. B. Fine, The Euclidean Bianchi groups, Comm. Algebra 18 (1990), no. 8, 2461--2484; MR
  117. C. R. Fletcher, Euclidean Rings, J. London Math. Soc. 4 (1971), 79--82
  118. G. Fontené, Sur les entiers algébriques de la forme $x+y\sqrt{-5}$, Nouv. ann. math. (4) 3 (1903), 209--214
  119. A. Förster, Zur Theorie einseitig euklidischer Ringe, Wiss. Z. Pädagog. Hochsch. "Karl Liebknecht" Potsdam 32 (1988), no. 1, 143--147 MR
  120. A. Förster, Zur Theorie einseitig euklidischer Ringe. II, Wiss. Z. Pädagog. Hochsch. "Karl Liebknecht" Potsdam 33 (1989), no. 1, 87--90 MR
  121. A. Förster, Zur Theorie einseitig euklidischer Ringe. III, Wiss. Z. Pädagog. Hochsch. "Karl Liebknecht" Potsdam 34 (1990), no. 1, 123--128; MR
  122. J. Fox, Finiteness of the number of quadratic fields with even discriminant and Euclidean algorithm, Ph.D. Thesis, Yale University 1935; see also Bull. Amer. Math. Soc. 41 (1935), p. 186
  123. C. F. Gauss, Disquisitiones Arithmeticae, 1801
  124. C. F. Gauss, Theoria residuorum biquadraticorum II, Werke II (1876), 93--148
  125. C. F. Gauss, Zur Theorie der komplexen Zahlen, Werke II (1876), 387--398
  126. F. A. W. George, Using the Euclidean algorithm to identify principal ideals and their generators in imaginary quadratic fields, manuscript 1994
  127. H. J. Godwin, On the inhomogeneous minima of certain norm-forms, J. London Math. Soc. 30 (1955), 114--119
  128. H. J. Godwin, On a conjecture of Barnes and Swinnerton-Dyer, Proc. Cambridge Phil. Soc. 59 (1963), 519--522
  129. H. J. Godwin, On Euclid's algorithm in some quartic and quintic fields, J. London Math. Soc. 40 (1965), 699--704
  130. H. J. Godwin, On the inhomogeneous minima of totally real cubic norm-forms, J. London Math. Soc. 40 (1965), 623--627
  131. H. J. Godwin, On Euclid's algorithm in some cubic fields with signature one, Quart. J. Math. Oxford 18 (1967), 333--338
  132. H. J. Godwin, Computations relating to cubic fields, Computers in number theory, London (1971), 225--229
  133. H. J. Godwin, J. R. Smith, On the Euclidean nature of four cyclic cubic fields, Math. Comp. 60 (1993), 421--423
  134. D. S. Gorskov, On the Euclidean algorithm in real quadratic fields (Russ.), Ucenye Zapisky Kazan. Univ. 101 (1941), 31--37
  135. D. S. Gorskov, Real quadratic fields without an Euclidean algorithm (Russ.), Ucenye Zapisky Kazan. Univ. 101 (1941), 37--42
  136. R. Gupta, M. Murty, V. Murty, The Euclidean algorithm for S-integers, Canad. Math. Soc. Conference Proc. 7 (1987), 189--201
  137. B. Hainke, Reellquadratische Zahlkörper mit euklidischem Algorithmen, diploma thesis Univ. Mainz, 1997
  138. M. Harper, McGill Univ. thesis 1997
  139. M. Harper, A proof that $Z[\sqrt{14}]$ os a Euclidean Domain, Ph. D. thesis McGill July 2000
  140. H. Hasse, Über eindeutige Zerlegung in Primelemente oder in Primhauptideale in Integri\-täts\-bereichen, J. Reine Angew. Math. 159 (1928), 3--12
  141. H. Heilbronn, On Euclid's algorithm in real quadratic fields, Proc. Cambridge Phil. Soc. 34 (1938), 521--526
  142. H. Heilbronn, On Euclid's algorithm in cubic self-conjugate fields, Proc. Cambridge Phil. Soc. 46 (1950), 377--382
  143. H. Heilbronn, On Euclid's algorithm in cyclic fields, Canad. J. Math. 3 (1950), 257--286; see also The collected papers of H. Heilbronn (1988)
  144. Heinhold, Verallgemeinerung und Verschärfung eines Minkowskischen Satzes Math. Z. 44 (1939), 659--688; ibid. 45 (1939), 176--184
  145. M. D. Hendy, Euclid and the fundamental theorem of arithmetic, Hist. Math. 2 (1975), 189--191
  146. D. Hensley, The number of steps in the Euclidean algorithm, J. Number Theory 49 (1994), no. 2, 142--182
  147. J. J. Hiblot, Des anneaux euclidiens dont le plus petit algorithme n'est pas à valeurs finies, C.R. Acad. Sci. Paris, Sér A 281 (1975), 411--414
  148. N. Hofreiter, Quadratische Zahlkörper ohne Euklidischen Algorithmus, Math. Ann. 110 (1935), 194-196
  149. N. Hofreiter, Quadratische Zahlkörper mit und ohne Euklidischen Algorithmus, Monatsh. Math. Phys. 42 (1935), 397--400
  150. L. K. Hua, On the distribution of quadratic non-residues and the Euclidean algorithm in real quadratic fields I, Trans. Amer. Math. Soc. 56 (1944), 537--546
  151. L. K. Hua, S.H. Min, On the distribution of quadratic non-residues and the Euclidean algorithm in real quadratic fields II, Trans. Amer. Math. Soc. 56 (1944), 547--569
  152. L. K. Hua, W. T. Shih, On the lack of the Euclidean algorithm in $R(\sqrt{61})$, Amer. J. Math. 67 (1945), 209--211
  153. A. Hurwitz, Über die Entwicklung komplexer Grössen in Kettenbrüche, Acta Math. 11 (1887), 187--200
  154. A. Hurwitz, Letter to L. Bianchi, 24.06.1895, Opere di L. Bianchi XI, 103--105
  155. A. Hurwitz, Zur Theorie der algebraischen Zahlen, Nachrichten von der k. Gesellschaft der Wissenschaften zu Göttingen, Math. -phys. Klasse, (1895), 324--331; see also Math. Werke 2 (1932), 236--243
  156. A. Hurwitz, Die unimodularen Substitutionen in einem algebraischen Zahlenkörper, Nachrichten von der k. Gesellschaft der Wissenschaften zu Göttingen, Math. -phys. Klasse, (1895), 332-356; see also Math. Werke 2 (1932), 244--268
  157. A. Hurwitz, Der Euklidische Divisionssatz in einem endlichen algebraischen Zahlkörper, Math. Zeitschrift 3 (1919), 123--126; see also Math. Werke 2 (1932), 471--474
  158. M. Ikeda, On the Euclidean kernel, Turk. J. Math. 17 (1993), 161--170; Zbl
  159. K. Inkeri, Über den Euklidischen Algorithmus in quadratischen Zahlkör\-pern, Ann. Acad. Sci. Fenn. Ser. A 41 (1947), 1--35
  160. K. Inkeri, Neue Beweise für einige Sätze zum Euklidischen Algorithmus in quadratischen Zahlkör\-pern, Ann. Univ. Turku. A IX (1948), 1--15
  161. K. Inkeri, Non-homogeneous binary quadratic forms, Den 11te Skandinaviske Matematiker Kongress Trondheim (1949), 216--224
  162. K. Inkeri, V. Ennola, The Minkowski constant for certain binary quadratic forms, Ann. Univ. Turku. Ser A I 25 (1957), 3--18
  163. D. H. Johnson, C. S. Queens, A. N. Sevilla, Euclidean real quadratic number fields, Arch. Math. 44 (1985), 344--368
  164. J. Mandavid, Zahlkörper höheren Grades mit Euklidischem Algorithmus, diploma thesis Univ. Mainz, 1998
  165. R. G. McKenzie, The ring of cyclotomic integers of modulus thirteen is norm-euclidean, Ph.D. thesis, Michigan State University 1988
  166. G. Kacerovsky, Der euklidische Algorithmus in einigen biquadratischen Zahlkörpern, Diss. Wien 1977
  167. G. Kalajdzic, On Euclidean algorithms with some particular properties, Duro Kurepa memorial volume. Publ. Inst. Math. (Beograd) (N.S.) 57 (71) (1995), 124--134; Zbl
  168. E. Kaltofen, H. Rolletschek, Computing greatest common divisors and factorizations in quadratic number fields, Math. Comp. 53 (1989), no. 188, 697--720
  169. A. Kanemitsu, K. Yoshiba, Euclidean rings, Bull. Fac. Sci. Ibaraki Univ. Math. 18 (1986), 1--5; MR
  170. R. P. Kelisky, Concerning the Euclidean algorithm, Fibonacci Quart. 3 (1965), 219--223; MR
  171. H. Kilian, Zur mittleren Anzahl von Schritten beim euklidischen Algorithmus, Elem. Math. 38 (1983), no. 1, 11--15
  172. B. Kiraly, G. Orosz, On a Euclidean ring (Hungarian), Acta Acad. Paedagog. Agriensis, Sect. Mat. (N.S.) 25 (1998), 71-76
  173. A. Knopfmacher, Elementary properties of the subtractive Euclidean algorithm, Fibonacci Quart. 30 (1992), no. 1, 80--83.
  174. A. Knopfmacher, J. Knopfmacher, The exact length of the Euclidean algorithm in Fq[X], Mathematika 35 (1988), no. 2, 297--304; Zbl
  175. A. Knopfmacher, J. Knopfmacher, Maximum length of the Euclidean algorithm and continued fractions in F(x), Applications of Fibonacci numbers, Vol. 3 (Pisa, 1988), 217--222, Kluwer Acad. Publ., Dordrecht, 1990
  176. A. Knopfmacher, J. Knopfmacher, The number of steps in the Euclidean algorithm over complex quadratic fields, BIT 31 (1991), no. 2, 286--292
  177. W. Knorr, Problems in the interpretation of Greek number theory; Euclid and the fundamental theorem of arithmetic, Studies in Hist. and Phil. Sci. 7 (1976), 353--368
  178. Ch. Kolb, Der euklidische Algorithmus in Körpern $\Q(\sqrt m\,)$, Kap. 3 der Zulassungsarbeit, Univ. Heidelberg 1976
  179. G. Kumar, Quadratic Euclidean domains, Math. Ed. 26 (1992), 180--185
  180. E. E. Kummer, Letters to L. Kronecker, 2.10.1844 and 16.10.1844, Coll. Papers I (1975), 87--92
  181. R. B. Lakein, Euclid's algorithm in complex quartic fields, Acta Arithm. 20 (1972), 393--400
  182. E. Landau, Vorlesungen über Zahlentheorie, Leipzig 1927
  183. D. Lazard, On the minimal algorithm in rings of imaginary quadratic integers, J. Number Theory 15 (1982), no. 2, 143--148
  184. F. Lemmermeyer, Euklidische Ringe, Diplomarbeit, Heidelberg 1989
  185. F. Lemmermeyer, The Euclidean algorithm in algebraic number fields, Expo. Math. 13, No. 5 (1995), 385-416
  186. F. Lemmermeyer, Gauss bounds of quadratic extensions, Publ. Math. Debrecen 50 (1997), 365--368
  187. F. Lemmermeyer, Euclid's algorithm in quartic CM-fields, in preparation
  188. V. Lemmlein, On Euclidean rings and rings of principal ideals, Doklady kad. Nauk SSSR (NS) 97 (1954), 585--587
  189. H. W. Lenstra, Lectures on Euclidean rings, Bielefeld 1974
  190. H. W. Lenstra, Euclid's algorithm in cyclotomic fields, Report 74-01, Dept. Math. Univ. Amsterdam (1974), 13pp
  191. H. W. Lenstra, Euclid's algorithm in cyclotomic fields, J. London Math. Soc. 10 (1975), 457--465
  192. H. W. Lenstra, Euclid's algorithm in cyclotomic fields, Tagungsbericht 33/75, Math. Forschungs\-institut Oberwolfach 1975
  193. H. W. Lenstra, Euclidean number fields of large degree, Invent. Math. 38 (1977), 237--254
  194. H. W. Lenstra, On Artin's conjecture and Euclid's algorithm in global fields, Invent. Math. 42 (1977), 201--224
  195. H. W. Lenstra, Quelques examples d'anneaux euclidiens, C.R. Acad. Sci. Paris Ser. A 289 (1978), 683--685
  196. H. W. Lenstra, Euclidean ideal classes, Journées Arithmetiques de Luminy, Asterisque 61 (1979), 121--131
  197. H. W. Lenstra, Euclidean number fields 1, Math. Intell. 2 (1979), 6--15
  198. H. W. Lenstra, Euclidean number fields 2, Math. Intell. 2 (1980), 73-77
  199. H. W. Lenstra, Euclidean number fields 3, Math. Intell. 2 (1980), 99--103
  200. A. Lepschy, G. A. Mian, U. Viaro, Euclid-type algorithm and its applications, Internat. J. Systems Sci. 20 (1989), no. 6, 945--956
  201. A. Leutbecher, Euklidischer Algorithmus und die Gruppe GL(2), Math. Ann. 231 (1977/78), 269--285
  202. A. Leutbecher, Euclidean number fields having a large Lenstra constant, Ann. Inst. Fourier 35 (1985), 83--106
  203. A. Leutbecher, Ausnahmeeinheiten und euklidische Zahlkörper, Tagungsbericht 35/86, Math. For\-schungs\-institut Oberwolfach 1986
  204. A. Leutbecher, New Euclidean fields by Lenstra's method of exceptional units, Proc. Conf. Number Theory and Arithm. Geometry Essen 1991; preprint 18/1991, Universität Essen, 79--80
  205. A. Leutbecher, J. Martinet, Constante de Lenstra et corps de nombres Euclidiens, Sémin. Théorie des nombres Univ. Bordeaux 1981/82 exp. no 4
  206. A. Leutbecher, J. Martinet, Lenstra's constant and Euclidean number fields, Journées Arithmetiques Metz, Astérisque 94 1982
  207. A. Leutbecher, G. Niklasch, On cliques of exceptional units and Lenstra's construction of Euclidean number fields, Number Theory Ulm 1987, Lecture Notes Math. 1380 (1989), 150--178
  208. F. J. van der Linden, Euclidean number rings with two infinite primes, Tagungsbericht Math. Forschungsinstitut Oberwolfach 35/81 1981
  209. F. J. van der Linden, Euclidean rings of integers of fourth degree fields, Number theory Noordwijkerhout, Lecture notes in Math. 1068 (1983), 139--148
  210. F. J. van der Linden, Euclidean rings with two infinite primes, Ph.D. thesis, Univ. Amsterdam 1984
  211. F. J. van der Linden, Euclidean rings with two infinite primes, CWI Tract 15 Centrum voor Wiskunde en Informatica 1985
  212. S. Lubelsky, Algorytm Euklidesa, Wiadom Mat. 42 (1937), 5--67
  213. S. Lubelsky, Unpublished results on number theory. I. Quadratic forms in a Euclidean ring, Acta Arith. 6 (1960/61), 217--224
  214. F. Lucius, Ringe mit einer Theorie des größten gemeinsamen Teilers, Mathematica Gottingensis 7 (1996); Zbl
  215. M.L. Madan, C.S. Queen, Euclidean function fields, J. Reine Angew. Math. 262/263 (1973), 271--277
  216. R. Markanda, Euclidean rings of algebraic numbers and functions, J. Algebra 37 (1975), no. 3, 425--446
  217. R. Markanda, V. S. Albis Gonzales, Euclidean algorithm in principal arithmetic algebras, Tamkang J. Math. 15 (1984), no. 2, 193--196
  218. J. M. Masley, On cyclotomic fields Euclidean for the norm map, Notices Amer. Math. Soc. 19 (1972), A-813; Abstr. \# 700 A-3
  219. J. M. Masley, On Euclidean rings of integers in cyclotomic fields, J. Reine Angew. Math. 272 (1975), 45--48
  220. J. F. Mestre, Corps Euclidiens, unites exceptionelles et courbes elliptiques, J. Number Theory 13 (1981), 123--137
  221. W.F. Meyer, Anwendung des erweiterten Euklid'schen Algorithmus auf Resultantenbildungen, Jahresber. DMV 16 (1907), 16--35
  222. S. H. Min, On the Euclidean algorithm in real quadratic fields, J. London Math. Soc. 22 (1947), 88--90
  223. S. H. Min, Euclidean algorithm in real quadratic fields, Sci. Rep. Nat. Tsing Hua Univ. Ser. A 5 (1948), 190--223
  224. T. E. Moore, On the least absolute remainder Euclidean algorithm, Fibonacci Quart. 30 (1992), no. 2, 161--165.
  225. T. Motzkin, The Euclidean algorithm, Bull. Am. Math. Soc. 55 (1949), 1142--1146
  226. M. Nagata, On Euclid algorithm, C. P. Ramanujan, A tribute, Tata Inst. Fund. Res. Stud. Math. 8 (1978), 175--186
  227. M. Nagata, Some remarks on Euclid rings, J. Math. Kyoto Univ. 25 (1985), no. 3, 421--422
  228. M. Nagata, On the definition of a Euclid ring, Adv. Stud. Pure Math. 11 (1987), 167--171; Zbl
  229. M. Nagata, A pairwise algorithm and its application to $\Z[\sqrt{14}]$, Algebraic geometry Seminar, Singapore (1987), 69--74
  230. M. Nagata, Pairwise algorithms and Euclid algorithms, Collection of papers dedicated to Prof. Jong Geun Park on his sixtieth birthday (Korean), 1--9, Jeonbug, Seoul, 1989;
  231. M. Nagata, Some questions on $\Z[\sqrt{14}\,]$, Algebraic geometry and its applications (Ch. Bajaj ed.), Conf. Purdue Univ. USA, June 1--4, 1990, Springer Verlag (1994), 327--332; Zbl
  232. W. Narkiewicz, Book review ``Collected Papers of H. Heilbronn'', DMV Jahresbericht 93 (1991), 4--5
  233. G. Niklasch, Ausnahmeeinheiten und Euklidische Zahlkörper, Diplomarbeit TU München 1986
  234. G. Niklasch, On Clarks example of a Euclidean field which is not norm-euclidean, manuscripta math. 83 (1994), 443--446; Zbl
  235. G. Niklasch, Family Portraits of exceptional units, to appear
  236. G. Niklasch, R. Qu\^{e}me, An improvement of Lenstra's criterion for euclidean number fields: The totally real case, Acta Arith. 58 (1991), 157--168
  237. P. Noordzij, Über das Produkt von vier reellen homogenen linearen Formen, Monatsh. Math. 71 (1967), 436--445
  238. G. H. Norton, On the asymptotic analysis of the Euclidean algorithm, J. Symbolic Comput. 10 (1990), no. 1, 53--58
  239. T. Ojala, Euclid's algorithm in the cyclotomic fields $\Q(\zeta_{16})$, Math. Comp. 31 (1977), 268--273
  240. O. T. O'Meara, On the finite generation of linear groups over Hasse domains, J. Reine Angew. Math. 217 (1965), 79--108
  241. A. Oneto, V. Ramirez, Non-Euclidean principal domains (Span.), Divulg. Mat. 1 (1993), 55--65; Zbl
  242. A. Oppenheim, Quadratic fields with and without Euclid's algorithm, Math. Ann. 109 (1934), 349--352
  243. H. Ostmann, Euklidische Ringe mit eindeutiger Partialbruchzerlegung, J. Reine Angew. Math. 188 (1950), 150--161
  244. J. Ouspensky, Note sur les nombres entiers dependant d'une racine cinquième de l'unité, Math. Ann. 66 (1909), 109-112; see also Moskau Math. Samml. 26, 1--17
  245. P. S. Papkov, Der Euklidische Algorithmus im quadratischen Zahlkörper mit beliebiger Klassenzahl, (Russ., German summary) \OE uvres sci. Univ. Etat, Rostoff sur Don 1 (1934), 15--60; Zbl
  246. P. S. Papkov, Über eine Anwendung des Euklidischen Algorithmus im quadratischen Zahlkörper mit beliebiger Klassenzahl (Russ., German summary), \OE uvres sci. Univ. Etat, Rost. s, Don 1 (1934), 61--78; Zbl
  247. F. Paulsen, B. Gordon, On the parity of some quantities related to the Euclidean algorithm, Amer. Math. Monthly 68 (1961), 900--901
  248. O. Perron, Quadratische Zahlkörper mit Euklidischem Algorithmus, Math. Ann. 107 (1932), 489--495
  249. G. Philibert, L'algorithme d'Euclide dans les corps cyclotomiques, Sé\-mi\-nai\-re d'Arith\-me\-tique, Saint-Etienne 1990-91-92, exp. no. 5, 59--67
  250. G. Picavet, Caracterisation de certains types d'anneaux euclidiens, Enseignement Math. 18 (1972), 245--254
  251. G. Pollak, On types of Euclidean norms (Russ.), Acta Sci. Math. Szeged 20 (1959), 252--268
  252. A. Popescu, On the Euclideanity in rings, Rev. Roumaine Math. Pures Appl. 27 (1982), no. 2, 181--185
  253. A.V. Prasad, A non-homogeneous inequality for integers in a special cubic field I, II, Indag. Math. 11 (1949), 55--65, 112--124
  254. C. S. Queen, Arithmetic Euclidean rings, Acta Arith. 26 (1972), 105--113
  255. R. Qu\^eme, A computer algorithm for finding new euclidean number fields, J. Théor. Nombres Bordeaux 10 (1998), 33-48; Zbl
  256. A.M. Rahimi, Rings with an almost division algorithm, Libertas Math. 13 (1993), 41-46; Zbl
  257. L. Rédei, Über den Euklidischen Algorithmus in reellquadratischen Zahlkörpern, J. Reine Angew. Math. 183 (1941), 183--192
  258. L. Rédei, Zur Frage des Euklidischen Algorithmus in quadratischen Zahlkörpern, Math. Ann. 118 (1942), 588--608
  259. R. Remak, Verallgemeinerung eines Minkowskischen Satzes. I. II., Math. Zeitschrift 17, 1-34, 18, 173-200 (1923). ??
  260. R. Remak, Über den Euklidischen Algorithmus in reell-quadratischen Zahlkörpern, Jahresber. DMV 44 (1934), 238--250
  261. G. Renault, Anneaux principaux et anneaux euclidiens, Gaz. Sci. Math. Quebec 1 (1977), 16--22
  262. F. Rivero, Anillos con algoritmo débil (Span.), Notas de Matem\'atica 49, Universidad de los Andes, 1982. ii+114 pp.
  263. K. A. Rodosskij, On Euclidean rings, Sov. Math. Dokl. 22 (1980), 186--189
  264. K. A. Rodosskij, The Euclidean algorithm (Russ.), Moskow 1988, 240 pp. ISBN: 5-02-013726-X
  265. H. Rolletschek, The Euclidean algorithm for Gaussian integers, EUROCAL'83, Lecture Notes Comp. Sci. 162 (1983), 12--23
  266. H. Rolletschek, On the number of divisions of the Euclidean algorithm applied to Gaussian integers, J. Symbolic Comput. 2 (1986), no. 3, 261--291
  267. H. Rolletschek, Shortest division chains in imaginary quadratic number fields, Symbolic and algebraic computation (Rome, 1988), 231--243, Lecture Notes in Comput. Sci., 358, Springer, Berlin, 1989
  268. H. Rolletschek, Shortest division chains in imaginary quadratic number fields, J. Symbolic Comp. 9 (1990), 321--354; Zbl
  269. B. Rosser, A generalization of the Euclidean algorithm to several dimensions, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 309--311
  270. B. Rosser, A generalization of the Euclidean algorithm to several dimensions, Duke Math. J. 9 (1942), 59--95
  271. P. A. Samet, The product of non-homogeneous linear forms I, Proc. Cambridge Phil. Soc. 50 (1954), 372--379
  272. P. A. Samet, The product of non-homogeneous linear forms II, Proc. Cambridge Phil. Soc. 50 (1954), 380--390
  273. P. Samuel, About Euclidean rings, J. Algebra 19 (1971), 282--301
  274. J. Sauvageot, Algorithmes d'Euclide dans certains corps biquadratiques, Sém. Delange-Pisot-Poitou, exp. no 15, 1972/73, 3pp
  275. J. Schatunowsky, Der grösste gemeinschaftliche Teiler von algebraischen Zahlen zweiter Ordnung mit negativer Diskriminante und die Zerlegung dieser Zahlen in Primfaktoren, Diss. Strassburg (1911), Leipzig 1912
  276. P. Schreiber, A supplement to J. Shallit's paper: "Origins of the analysis of the Euclidean algorithm", Historia Math. 22 (1995), no. 4, 422--424
  277. R. Schroeppel, $\Q(\root 5 \of 2)$ is norm-Euclidean, unpublished, 1993
  278. V. Schulze, Verallgemeinerte euklidische Algorithmen, Arch. Math. 64 (1995), 313--315
  279. L. Schuster, Reellquadratische Zahlkörper ohne Euklidischen Algorithmus, Monatsh. Math. Phys. 47 (1938), 117--127
  280. J. Shallit, Origins of the analysis of the Euclidean algorithm, Historia Math. 21 (1994), no. 4, 401--419; MR; Zbl
  281. D. B. Shapiro, R. K. Markanda, E. Brown, Some Euclidean properties for real quadratic fields, Acta Arith. 47 (1986), 143--152
  282. J. R. Smith, On Euclid's algorithm in some cyclic cubic fields, J. London Math. Soc. 44 (1969), 577--582
  283. J. R. Smith, The inhomogeneous minima of some totally real cubic fields, Computers in number theory, London (1971), 223--224
  284. H. M. Stark, The papers on Euclid's algorithm, The Collected papers of H. Heilbronn (1990), 586--587
  285. H. P. F. Swinnerton-Dyer, The inhomogeneous minima of complex cubic norm forms, Proc. Cambridge Phil. Soc. 50 (1954), 209--219
  286. Y. Tanimura, Euclidean algorithm in quadratic fields II, Sci. Rep. Lib. Arts Ed. Gifu Univ. Natur. Sci. 3 (1964/65), 339-345
  287. Y. Tanimura, Euclidean algorithm in quadratic fields III, Sci. Rep. Lib. Arts Ed. Gifu Univ. Natur. Sci. 4 (1967), 13--18
  288. Y. Tanimura, Euclidean algorithm in cubic fields I, Sci. Rep. Lib. Arts Ed. Gifu Univ. Natur. Sci. 4 (1969), 135--137
  289. Y. Tanimura, Non Euclidean point in field $\Q(\sqrt{97})$, Bull. Tokai Women's College 2 (1982), 39--44
  290. H. Tapia-Recillos, J. Valle Can, Non-Euclidean principal ideal domains of algebraic integers (Span.), Proceedings of the XIX-th National Congress of the Mexican Math. Soc. 1 (1986), 231--252
  291. E. M. Taylor, Euclid's algorithm in cubic fields with complex conjugates, Ph.D. thesis London (1975)
  292. E. M. Taylor, Euclid's algorithm in cubic fields with complex conjugates, J. London Math. Soc. 14 (1976), 49--54
  293. J. G. Tena Ayuso, Subannillos Euclideos de cuerpos quadraticos reales, Rev. Math. Hisp.-Amer. 37 (1977), 43--50
  294. C. Traub, Theorie der sechs einfachsten Systeme complexer Zahlen, I, II, Beilage zum Programm des Grossh. Lyceums, Mannheim 1867/1868
  295. S. Treatman, Euclidean Systems, Diss. Michigan, 1996
  296. P. Varnavides, Note on non-homogeneous quadratic forms, Quart. J. Math. Oxford 19 (1948), 54--58
  297. P. Varnavides, Non-homogeneous quadratic forms, Proc. Ned. Acad. Wet. 51 (1948), 396--404
  298. P. Varnavides, Euclid's algorithm in real quadratic fields, Prakt. Math. Athenon 24 (1949), 117--123
  299. P. Varnavides, On the quadratic form x2-7y2, Quart. J. Math. Oxford 20 (1949), 124--128
  300. P. Varnavides, The Euclidean real quadratic number fields, Indag. Math. 14 (1952), 111--122
  301. P. Varnavides, The nonhomogeneous minima of a class of binary quadratic forms, J. Number Theory 2 (1970), 333--341
  302. L. N. Vaserstein, On the group SL(2) for Dedekind domains of arithmetic type, Mat. Sb. 89 (131) (1972), 313--322
  303. E. M. Vechtomov, On the general theory of Euclidean rings (Russian), Abelian groups and modules, No. 9 (Russian), 3--7, 155, Tomsk. Gos. Univ., Tomsk, 1990; MR
  304. G. R. Veldkamp, Remark on Euclidean rings (Dutch), Nieuw Tijdschr. Wisk. 48 (1960/61), 268--270
  305. P. L. Wantzel, Notes sur la théorie des nombres complexes, Compt. Rendus de l'Academie des Sciences 24 (1847)
  306. P. L. Wantzel, Extraits des Proces-Verbaux des Seances, Soc. Philomatique de Paris (1848), 19--22
  307. J. H. M. Wedderburn, Non-commutative domains of integrity, J. Reine Angew. Math. 167 (1932), 129--141; Fd M 58.0148.01, Zbl
  308. P. J. Weinberger, On Euclidean rings of algebraic integers, Proc. Symp. Pure Math. 24 Analytic number theory, AMS, (1973), 321--332
  309. J. C. Wilson, A principal ideal ring that is not a Euclidean ring, Math. Mag. 46 (1973), 34--38; Zbl; see also Selected Papers on Algebra (R. Brink, ed.), 1977, p. 79--82
  310. K. C. Yang, Quadratic fields without Euclid's algorithm, Sci. Rep. Nat. Tsing-Hua Univ. A 3 (1935), 261--264
  311. H. You, Some notes on rings with Euclidean algorithms (Chin.), Dongbei Shida Xuebao 1984, no. 2, 17--19; Zbl
  312. T. Zaupper, A quasi-Euclidean algorithm (Hungar.), Bull. Appl. Math. 31, no. 210 (1983), 127--145
  313. Y. M. Zheng, The structure of Euclidean rings with unique division (Chin.), J. Math. (Wuhan) 1 (1981), no. 2, 185--188 Zbl
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