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Some facts concerning integral representations of ideals in an algebraic number field

  • Olga Taussky
Part I
Part of the Lecture Notes in Mathematics book series (LNM, volume 882)

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Bibliography

  1. Baumert, L.: Query 153. Notices Amer. Math. Soc. 25, 252 (1978).Google Scholar
  2. Bhandari, S. K.: Ideal matrices for Dedekind domains. J. Indian Math. So. 42, 109–126 (1978).MathSciNetzbMATHGoogle Scholar
  3. Bhandari, S. K. and Nanda, V. C.: Ideal matrices for relative extensions. Abh. Math. Sem. Univ. Hamburg 49, 3–17 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  4. Dade, E. C.: Invertible ideals capitulate sooner or later. Manuscript.Google Scholar
  5. Estes, D.: Determinants of Galois autormorphisms of maximal commutative rings of 2×2 matrices. Linear Alg. and Appl. 21 (1979), 225–243.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Foster, L.: On the characteristic roots of the product of certain rational integral matrices of order two. California Institute of Technology thesis, 1964.Google Scholar
  7. Gustafson, W. H.: Remarks on the history and applications of integral representations. These Springer Lecture NotesGoogle Scholar
  8. Guralnich, R.: Isomorphism of modules under ground ring extension. Lin. Alg. and Appl. to appear.Google Scholar
  9. Jacobinski, H.: Uber Geschlechter von Ordnungen. J. reine und angew. Math. 230, 29–39 (1968).MathSciNetzbMATHGoogle Scholar
  10. Kruglzak, S. A.: Precise ideals of integer matrix rings of the second order. Ukrain. Mat. Z. 18, 58–64 (1966).MathSciNetCrossRefGoogle Scholar
  11. Levy, L. S.: Almost diagonal matrices over Dedekind domains. Math. Zeitschr. 124, 89–99 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  12. MacDuffee, C. C.: The theory of matrices. Ergebnisse der Mathematik, Springer 1933.Google Scholar
  13. MacDuffee, C. C.: An introduction to the theory of ideals in linear associative rings. Trans. Amer. Math. Soc. 31, 71–90 (1928).MathSciNetCrossRefzbMATHGoogle Scholar
  14. MacDuffee, C. C.: A method for determining the canonical basis of an ideal in an algebraic field. Math. Ann. 105, 663–665 (1931).MathSciNetCrossRefzbMATHGoogle Scholar
  15. Magnus, W.: Non euclidean tesselations and their groups. Academic Press 1974.Google Scholar
  16. Mahler, K.: Inequalities for ideal bases in algebraic number fields. J. Austral. Math. Soc. 4, 425–448 (1964).MathSciNetCrossRefzbMATHGoogle Scholar
  17. Malysev, A. V. and Paceo, U. V.: On the arithmetic of second order matrices. Zap. Nauc. Ses., Moscow 93, 41–86 (1980).MathSciNetGoogle Scholar
  18. Mann, H. and Yamamoto, K.: On canonical bases of ideals. J. of Combinatorial Theory 2, 71–76 (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  19. Newman, M.: Integral matrices. Acad. Press 1972.Google Scholar
  20. Plesken, W.: Beiträge zur Bestimmung der endlichen irreduziblen Untergruppen von GL (n,Z) und ihrer ganzzahligen Darstellungen. Ph. D. Thesis, Aachen, 1–69 (1976).Google Scholar
  21. Plesken, W. and Pohst, M.: On maximal finite irreducible subgroups of GL (n,Z) II, The six dimensional case. Mathematics of Computation 31, 552–573 (1977).MathSciNetzbMATHGoogle Scholar
  22. Rademacher, H.: Zur Theorie der dedekindschen Summen. Math. Z. 63 445–463 (1955)MathSciNetCrossRefGoogle Scholar
  23. Rehm, H. P.: On Ochoa's special matrices in matrix classes. Linear Algebra and Appl. 17, 181–188 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  24. Rehm, H. P.: On a theorem of Gausz concerning the number of integral solutions of the equation x2+y2+z2=m. Seminar Notes on ternary forms and norms, to appear, Dekker.Google Scholar
  25. Reiner, I., Roggenkamp, K. W.: Integral representations. Lecture Notes in Mathematics, 744, Springer 1979Google Scholar
  26. Roggenkamp, K. W. and Huber-Dyson, V.: Lattices over orders. Lecture Notes in Mathematics 115, Springer 1970.Google Scholar
  27. Rosenbrock, H. H.: State-Space and Multivariable Theory. J. Wiley, 1970.Google Scholar
  28. Schur, I.: Über Ringbereiche im Gebiet der ganzzahligen linearen Substitutionen. Sitzg. Ber. Preuss. Akad. Wiss. (1922), 145–168.Google Scholar
  29. Siegel, C. L.: Über die analytische Theorie der quadratischen Formen III. Ann. Math. 38, 212–291 (1937).CrossRefzbMATHGoogle Scholar
  30. Taussky, O.: On a theorem of Latimer and MacDuffee. Canadian Journal Mathematics 1, 300–302 (1949).MathSciNetCrossRefzbMATHGoogle Scholar
  31. Taussky, O.: On matrix classes corresponding to an ideal and its inverse. Illinois Journal Mathematics 1, 103–113 (1957).MathSciNetzbMATHGoogle Scholar
  32. Taussky, O.: Ideal matrices, I. Archiv der Mathematik 13, 275–282 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Taussky, O.: Ideal matrices, II. Math. Ann. 150, 218–225 (1963).MathSciNetCrossRefzbMATHGoogle Scholar
  34. Taussky, O.: On the similarity transformation between an integral matrix with irreducible characteristic polynomial and its transpose. Math. Ann. 166, 60–63 (1966).MathSciNetCrossRefzbMATHGoogle Scholar
  35. Taussky, O.: Research Problem 10. Bull. Amer. Math. Soc. 64, 124 (1958).MathSciNetCrossRefGoogle Scholar
  36. Taussky, O.: Additive commutators of rational 2×2 matrices. Linear Alg. and Appl. 12 (1975), 1–6.MathSciNetCrossRefzbMATHGoogle Scholar
  37. Taussky, O.: Connections between algebraic number theory and integral matrices. Appendix to H. Cohn, A classical invitation to algebraic numbers and class fields, Springer (1978).Google Scholar
  38. Taussky, O.: A diophantine problem arising out of similarity classes of integral matrices. J. of Number Theory 11, 472–475 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  39. Taussky, O.: Some facts concerning integral representations of the ideals in an algebraic number field. Linear Alg. and Appl. 31, 245–248 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  40. Taussky, O.: Composition of binary integral quadratic forms and composition of matrix classes. To appear in Lin. and Multilin Algebra.Google Scholar
  41. Venkow, V.: On the arithmetic of quaternions. Bull. de l'académie des Sciences de l'URSS, 205–246; 489–504; 535–562; 607–622 (1922, 1929).Google Scholar
  42. Wagner, G. B.: Ideal matrices and ideal vectors. Math. Ann. 183, 241–249 (1969).MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag 1981

Authors and Affiliations

  • Olga Taussky
    • 1
  1. 1.California Institute of TechnologyPasadenaUSA

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