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dc.contributor.advisorWashington, Lawrenceen_US
dc.contributor.authorBlagg, Daviden_US
dc.date.accessioned2014-06-24T05:49:06Z
dc.date.available2014-06-24T05:49:06Z
dc.date.issued2014en_US
dc.identifier.urihttp://hdl.handle.net/1903/15215
dc.description.abstractLet F<sub>0</sub> = <bold>Q</bold>((-d)<super>&half;</super>), K<sub>0</sub> = <bold>Q</bold>(d<super>&half;</super>), and L<sub>0</sub> = <bold>Q</bold>(d<super>&half;</super>, i) with d a square-free positive integer such that 2 does not divide d. Let L<sub>j</sub> = L<sub>0</sub>(zeta<sub>2<super>2+j</super></sub>) so that the fields L<sub>j</sub> are the cyclotomic <bold>Z</bold><sub>2</sub>-extension of L<sub>0</sub>. We determine when fourth roots of certain elements of K<sub>0</sub> generate unramified extensions of L<sub>j</sub>. In particular, for elements of K<sub>0</sub> that are relatively prime to 2 and are generators of principal ideals that are fourth powers, we give explicit congruence conditions under which the fourth root of the element gives an unramified extension. For any such element gamma, we show that if there is some j such that L<sub>j</sub>(gamma<super>1/4</super>)/L<sub>j</sub> is unramified, then L<sub>2</sub>(gamma<super>1/4</super>)/L<sub>2</sub> is unramified. We also show that when (2) is split in F<sub>0</sub>, L<sub>2</sub>(gamma<super>1/4</super>)/L<sub>2</sub> is unramified for any such gamma. This result is analogous to a result by Hubbard and Washington in which they work with the cyclotomic <bold>Z</bold><sub>3</sub>-extension of <bold>Q</bold>((-d)<super>&half;</super>, zeta<sub>3</sub>) when 3 does not divide d and consider extensions generated by cube roots of elements in <bold>Q</bold>((3d)<super>&half;</super>). However, many more technical problems arise in the present work because the degree of the extension L<sub>j</sub>/K<sub>j</sub> is not relatively prime to the degrees of the extensions being generated. In order to prove our main results, we also give a congruence condition, which, for any number field K containing i and for any element gamma in K with gamma relatively prime to 2 and gamma a generator of an ideal that is a fourth power, dictates whether or not adjoining a fourth root of gamma to K gives an unramified extension.en_US
dc.language.isoenen_US
dc.titleUnramified Extensions of the Cyclotomic Z_2-Extension of Q(sqrt(d),i)en_US
dc.typeDissertationen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.contributor.departmentMathematicsen_US
dc.subject.pqcontrolledMathematicsen_US
dc.subject.pquncontrolledIwasawa theoryen_US
dc.subject.pquncontrolledquadratic fieldsen_US
dc.subject.pquncontrolledunramified extensionsen_US


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