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# On the Characteristic Roots of the Product of Certain Rational Integral Matrices of Order Two

## Citation

Foster, Lorraine Lois Turnbull (1964) On the Characteristic Roots of the Product of Certain Rational Integral Matrices of Order Two. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/JE7E-1393. https://resolver.caltech.edu/CaltechETD:etd-09172002-111103

## Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Let N(p,q) denote the companion matrix of x[superscript 2] + px + q, for rational integers p and q, and let M(p,q)=N(p,q)(N(p,q))'. Further let F(M(p,q)) and F(N(p,q)) denote the fields generated by the characteristic roots of M(p,q) and N(p,q) over the rational field, R. This thesis is concerned with F(M(p,q)), especially in relation to F(N(p,q)). The principal results obtained are outlined as follows: Let S be the set of square-free integers which are sums of two squares. Then F(M(p,q)) is of the form R[...], where c [...] S. Further, F(M(p,q)) = R if and only if pq = 0. Suppose c [...] S. Then there exist infinitely many distinct pairs of integers (p,q) such that F(M(p,q)) = R[...]. Further, if c [...] S., there exists a sequence {(p[subscript n],q[subscript n])} of distinct pairs of integers such that F(N(p[subscript n],q[subscript n])) =R[...], and F(MN(p[subscript n],q[subscript n])) = R[...], where the d[subscript n] are some integers such that c,d[subscript n] = 1. If c [...] S and c is odd or c = 2, there exists a sequence {(p'[subscript n],q'[subscript n])} of distinct pairs of integers such that F(M(p'[subscript n],q'[subscript n)) = R[...] and F(N(p'[subscript n],q'[subscript n)) = R[...], for some integers d'[subscript n] such that (c,d'[subscript n]) = 1. There are five known pairs of integers (p,q), with pq [not equalling] 0 and q [not equalling 1, such that F(M(p,q)) and F(N(p,q)) coincide. For q [...] and for certain odd integers q, the fields F(M(p,q)) and F(N(p,q)) cannot coincide for any integers p. Finally, for any integer p [not equalling] (or q [not equalling] 0, -1) there exist at most a finite number of integers q (or p) such that the two fields coincide.

Item Type: Thesis (Dissertation (Ph.D.)) (Mathematics and Philosophy) California Institute of Technology Physics, Mathematics and Astronomy Mathematics Philosophy Public (worldwide access) Taussky-Todd, Olga Unknown, Unknown 28 April 1964 CaltechETD:etd-09172002-111103 https://resolver.caltech.edu/CaltechETD:etd-09172002-111103 10.7907/JE7E-1393 No commercial reproduction, distribution, display or performance rights in this work are provided. 3570 CaltechTHESIS Imported from ETD-db 17 Sep 2002 19 Jan 2024 00:59

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