Citation
Foster, Lorraine Lois (1964) On the characteristics roots of the product of certain rational matrices of order two. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd09172002111103
Abstract
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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 squarefree 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.)) 

Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  28 April 1964 
Record Number:  CaltechETD:etd09172002111103 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd09172002111103 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  3570 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  17 Sep 2002 
Last Modified:  26 Dec 2012 03:01 
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