Indices of Principal Orders in Algebraic Number Fields

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.

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