On finding imaginary quadratic fields where the class group has higher p-rank

Los, Leendert (2018) On finding imaginary quadratic fields where the class group has higher p-rank. Master's Thesis / Essay, Mathematics.

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Abstract

In this thesis, we derive an isomorphism between some subgroup of the multiplicative group of an algebraic number field modulo m-th powers and the elements of the class group that have order dividing m. Using p-adic techniques it can be tested whether elements in this subgroup are different or even independent, providing a way to find independent elements of order dividing m in the class group. This can be used to find lower bounds on the q-rank of the class group (for primes q dividing m). In case of imaginary quadratic fields, a correspondence between cubic norm equations and ideal classes of order dividing 3 as used by D.A. Buell and D. Shanks and R. Serafin is studied. After making this correspondence precise, it is generalised to arbitrary odd m, and it is interpreted in terms of the isomorphism above. Furthermore, results of J.J. Solderitsch, M. Craig and Y. Yamamoto about independency of elements of specific order in the class group are interpreted as special cases of the independence-showing above. Finally, the isomorphism above is used to consider a relation between rational points on elliptic curves and elements of order dividing 3 in class groups of quadratic fields as described by M. van Beek.

Item Type:	Thesis (Master's Thesis / Essay)
Supervisor name:	Top, J. and Wubs, F.W.
Degree programme:	Mathematics
Thesis type:	Master's Thesis / Essay
Language:	English
Date Deposited:	31 Aug 2018
Last Modified:	10 Sep 2018 12:39
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/18493

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