Reduced Ideals and Periodic Sequences in Pure Cubic Fields

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The “infrastructure” of quadratic fields is a body of theory developed by Dan Shanks, Richard Mollin and others, in which they relate “reduced ideals” in the rings and sub-rings of integers in quadratic fields with periodicity in continued fraction expansions of quadratic numbers. In this thesis, we develop cubic analogs for several infrastructure theorems. We work in the field K=Q(), where 3=m for some square-free integer m, not congruent to ±1, modulo 9. First, we generalize the definition of a reduced ideal so that it applies to K, or to any number field. Then we show that K has only ... continued below

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iv, 44 pages

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Jacobs, G. Tony August 2015.

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  • Jacobs, G. Tony

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The “infrastructure” of quadratic fields is a body of theory developed by Dan Shanks, Richard Mollin and others, in which they relate “reduced ideals” in the rings and sub-rings of integers in quadratic fields with periodicity in continued fraction expansions of quadratic numbers. In this thesis, we develop cubic analogs for several infrastructure theorems. We work in the field K=Q(), where 3=m for some square-free integer m, not congruent to ±1, modulo 9. First, we generalize the definition of a reduced ideal so that it applies to K, or to any number field. Then we show that K has only finitely many reduced ideals, and provide an algorithm for listing them. Next, we define a sequence based on the number alpha that is periodic and corresponds to the finite set of reduced principal ideals in K. Using this rudimentary infrastructure, we are able to establish results about fundamental units and reduced ideals for some classes of pure cubic fields. We also introduce an application to Diophantine approximation, in which we present a 2-dimensional analog of the Lagrange value of a badly approximable number, and calculate some examples.

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iv, 44 pages

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  • August 2015

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  • March 4, 2016, 4:14 p.m.

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  • May 18, 2017, 8 a.m.

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Jacobs, G. Tony. Reduced Ideals and Periodic Sequences in Pure Cubic Fields, dissertation, August 2015; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc804842/: accessed October 23, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .