## Citation

Knight, Melvin John
(1975)
*Indices of Principal Orders in Algebraic Number Fields.*
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/hcvt-yx83.
https://resolver.caltech.edu/CaltechTHESIS:08262021-155312274

## Abstract

Let K be an extension of Q of degree n and D_{K} the ring of integers of K. If θ is an algebraic integer of K and K = Q(θ), then Z[θ] is a suborder of D_{K} of finite index. This index is called the index of θ. If k is a rational integer, the numbers θ and θ + k have equal indices. Define two numbers to be equivalent if their difference is a rational integer.

Using Schmidt's extension of Thue's Theorem it is shown that in any field of degree less than or equal to four there exist only a finite number of inequivalent numbers with index bounded by any given number. This is true for every finite extension of Q and a proof is given using a slight generalization of Schmidt's Theorem.

An application of Schmidt's Theorem to a problem on the units in a cyclic field of prime degree is given.

Item Type: | Thesis (Dissertation (Ph.D.)) | ||||||||
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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): | - Taussky-Todd, Olga
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Thesis Committee: | - Taussky-Todd, Olga (chair)
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Defense Date: | 16 May 1975 | ||||||||

Funders: |
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Record Number: | CaltechTHESIS:08262021-155312274 | ||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechTHESIS:08262021-155312274 | ||||||||

DOI: | 10.7907/hcvt-yx83 | ||||||||

Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||

ID Code: | 14342 | ||||||||

Collection: | CaltechTHESIS | ||||||||

Deposited By: | Benjamin Perez | ||||||||

Deposited On: | 30 Aug 2021 16:44 | ||||||||

Last Modified: | 30 Aug 2021 16:44 |

## Thesis Files

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- Final Version
See Usage Policy. 15MB |

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