Citation
Bender, Edward Anton (1966) Symmetric representation of an integral domain over a subdomain. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:09172015140824031
Abstract
Let F(θ) be a separable extension of degree n of a field F. Let Δ and D be integral domains with quotient fields F(θ) and F respectively. Assume that Δ ᴝ D. A mapping φ of Δ into the n x n D matrices is called a Δ/D rep if (i) it is a ring isomorphism and (ii) it maps d onto dI_{n} whenever d ϵ D. If the matrices are also symmetric, φ is a Δ/D symrep.
Every Δ/D rep can be extended uniquely to an F(θ)/F rep. This extension is completely determined by the image of θ. Two Δ/D reps are called equivalent if the images of θ differ by a D unimodular similarity. There is a onetoone correspondence between classes of Δ/D reps and classes of Δ ideals having an n element basis over D.
The condition that a given Δ/D rep class contain a Δ/D symrep can be phrased in various ways. Using these formulations it is possible to (i) bound the number of symreps in a given class, (ii) count the number of symreps if F is finite, (iii) establish the existence of an F(θ)/F symrep when n is odd, F is an algebraic number field, and F(θ) is totally real if F is formally real (for n = 3 see Sapiro, “Characteristic polynomials of symmetric matrices” Sibirsk. Mat. Ž. 3 (1962) pp. 280291), and (iv) study the case D = Z, the integers (see Taussky, “On matrix classes corresponding to an ideal and its inverse” Illinois J. Math. 1 (1957) pp. 108113 and Faddeev, “On the characteristic equations of rational symmetric matrices” Dokl. Akad. Nauk SSSR 58 (1947) pp. 753754).
The case D = Z and n = 2 is studied in detail. Let Δ’ be an integral domain also having quotient field F(θ) and such that Δ’ ᴝ Δ. Let φ be a Δ/Z symrep. A method is given for finding a Δ’/Z symrep ʘ such that the Δ’ ideal class corresponding to the class of ʘ is an extension to Δ’ of the Δ ideal class corresponding to the class of φ. The problem of finding all Δ/Z symreps equivalent to a given one is studied.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Mathematics  
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:  8 February 1966  
Funders: 
 
Record Number:  CaltechTHESIS:09172015140824031  
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:09172015140824031  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9157  
Collection:  CaltechTHESIS  
Deposited By:  Leslie Granillo  
Deposited On:  17 Sep 2015 22:05  
Last Modified:  17 Sep 2015 22:05 
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