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Type of Document	Dissertation
Author	Foster, Lorraine Lois
URN	etd-09172002-111103
Persistent URL	http://resolver.caltech.edu/CaltechETD:etd-09172002-111103
Title	On the characteristics roots of the product of certain rational matrices of order two
Degree	PhD
Option	Mathematics
Advisory Committee	Advisor Name Title Unknown Committee Member
Keywords	None
Date of Defense	1964-04-28
Availability	unrestricted
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.
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