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Indices of Principal Orders in Algebraic Number Fields


Knight, Melvin John (1975) Indices of Principal Orders in Algebraic Number Fields. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/hcvt-yx83.


Let K be an extension of Q of degree n and DK 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 DK 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.))
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
Thesis Committee:
  • Taussky-Todd, Olga (chair)
Defense Date:16 May 1975
Funding AgencyGrant Number
Record Number:CaltechTHESIS:08262021-155312274
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14342
Deposited By: Benjamin Perez
Deposited On:30 Aug 2021 16:44
Last Modified:30 Aug 2021 16:44

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