## Departmental PhD Thesis Exam – Zackary Wolske

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Monday, June 18, 2018
2:10 p.m.
BA6183

PhD Candidate:  Zackary Wolske
Supervisor:   Henry Kim
Thesis title:   Number Fields with Large Minimal Index

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Abstract:

The index of an integral element alpha in a number field K with discriminant D_K is the index of the subring Z[alpha] in the ting of integers O_K. The minimal index m(K) is taken over all alpha in O_K that generate the field. This thesis proves results of the form m(K) << |D_K|^U for all Galois quartic fields and composites of totally real Galois fields with imaginary quadratic fields, and of the form m(K) >> |D_K|^L for infinitely many pure cubic fields, both types of Galois quartic fields, and the same composite fields, with U and L depending only on the type of field. The upper bounds are given by explicit elements and depend on finding a factorization of the index form, while the lower bounds are established via effective Diophantine approximation, minima of binary quadratic forms, or norm inequalities. The upper bounds improve upon known results, while the lower bounds are entirely new. In the case of imaginary biquadratic quartic fields and the composite fields under consideration, the upper and lower bounds match.

A copy of the thesis can be found here:  ZWolskePhDThesisJune14