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On a Theorem of Latimer and Macduffee

Published online by Cambridge University Press:  20 November 2018

Olga Taussky
Olga Taussky*
Affiliation:
Institute for Numerical Analysis, National Bureau of Standards
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 3 , 01 June 1949 , pp. 300 - 302
Copyright
Copyright © Canadian Mathematical Society 1949

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