On a Theorem of Latimer and Macduffee

Olga Taussky

doi:10.4153/CJM-1949-026-1

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The matrix solutions of an irreducible algebraic equation with integral coefficients were studied by Latimer and MacDuffee. They considered matrices with rational integers as elements. If A is such a matrix, then all matrices of the “class” S1 AS will again be solutions if S is a matrix of determinant ± 1. On the other hand, in general all solutions cannot be derived in this way from one solution only. It was in fact shown that the number of classes of matrix solutions coincides with the number of different classes of ideals in the ring generated by an algebraic root of the same equation.

References

1 Latimer, C. G. and MacDuffee, C. C., “A Correspondence Between Classes of Ideals and Classes of Matrices,” Ann. of Math., vol. 34 (1933), 313-316.Google Scholar See also related work in Speiser, A., Theorie der Gruppen (Springer, 1937) ;Google Scholar B. van der Waerden, L., Gruppen von linearen Transformationen (Springer, 1935);Google Scholar Zassenhaus, H., “Neuer Beweisder Endlichkeit der Klassenzahl bei unimodularer Äquivalenz endlicher ganzzahliger Substitutionsgruppen,” Abh. Math. Sem. Hansischen Univ., vol. 12 (1938), 276–288.Google Scholar

2 See e.g. Bambah, R. P. and Chowla, S., “On Integer Roots of the Unit Matrix,” Proc. Nat. Inst. Sci. India, vol. 13 (1937), 241–246.Google Scholar

3 See e.g. Wedderburn, J. H. M., “Lectures on Matrices,” Amer, Math. Soc. Colloquium Publications, vol. 17 (1934), 27.Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Rademacher, Hans 1955. Zur Theorie der Dedekindschen Summen. Mathematische Zeitschrift, Vol. 63, Issue. 1, p. 445.

Reiner, Irving 1957. Integral representations of cyclic groups of prime order. Proceedings of the American Mathematical Society, Vol. 8, Issue. 1, p. 142.

Rapaport, Elvira Strasser 1959. Note on Nielsen transformations. Proceedings of the American Mathematical Society, Vol. 10, Issue. 2, p. 228.

Taussky, Olga 1960. Matrices of rational integers. Bulletin of the American Mathematical Society, Vol. 66, Issue. 5, p. 327.

Taussky, Olga 1962. Ideal matrices. I. Archiv der Mathematik, Vol. 13, Issue. 1, p. 275.

Taussky, Olga 1966. On the similarity transformation between an integral matrix with irreducible characteristic polynomial and its transpose. Mathematische Annalen, Vol. 166, Issue. 1, p. 60.

Reiner, Irving 1970. A survey of integral representation theory. Bulletin of the American Mathematical Society, Vol. 76, Issue. 2, p. 159.

1972. Vol. 45, Issue. , p. 216.

CrossRef

1974. Noneuclidean Tesselations and Their Groups. Vol. 61, Issue. , p. 199.

Taussky, Olga 1974. A result concerning classes of matrices. Journal of Number Theory, Vol. 6, Issue. 1, p. 64.

Plesken, Wilhelm and Pohst, Michael 1977. On maximal finite irreducible subgroups of 𝐺𝐿(𝑛,𝑍). II. The six dimensional case. Mathematics of Computation, Vol. 31, Issue. 138, p. 552.

Rehm, Hans P. 1977. On Ochoa's special matrices in matrix classes. Linear Algebra and its Applications, Vol. 17, Issue. 2, p. 181.

Taussky, Olga 1979. A diophantine problem arising out of similarity classes of integral matrices. Journal of Number Theory, Vol. 11, Issue. 3, p. 472.

Taussky, Olga 1981. Integral Representations and Applications. Vol. 882, Issue. , p. 145.

Taussky, Olga 1982. The many aspects of the Pythagorean triangles. Linear Algebra and its Applications, Vol. 43, Issue. , p. 285.

Midorikawa, Hisaichi 1982. On regular elliptic conjugacy classes of the Siegel modular group. Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 58, Issue. 3,

Baker, Kirby A. 1983. Strong shift equivalence of 2 × 2 matrices of non-negative integers. Ergodic Theory and Dynamical Systems, Vol. 3, Issue. 4, p. 501.

Taussky, Olga 1983. Emmy Noether in Bryn Mawr. p. 47.

Estes, Dennis R and Guralnick, Robert M 1984. Representations under ring extensions: Latimer-MacDuffee and Taussky correspondences. Advances in Mathematics, Vol. 54, Issue. 3, p. 302.

Wallace, D. I. 1984. Conjugacy classes of hyperbolic matrices in 𝑆𝑙(𝑛,𝑍) and ideal classes in an order. Transactions of the American Mathematical Society, Vol. 283, Issue. 1, p. 177.

Download full list

Article contents

On a Theorem of Latimer and Macduffee

Extract

References

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests