טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentJoseph Cohen
SubjectPrimitive Roots in Algebraic Number Fields
DepartmentDepartment of Mathematics
Supervisors Full Professor Sonn Jack
Full Professor Zeev Rudnick


Abstract

We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields.  Given such a nontrivial unit, for a rational prime  which is inert in the field the maximal order of this unit modulo  is . An extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. This is known at present only under the Generalized Riemann Hypothesis. Unconditionally, we show that, for any choice of 7 units in different real quadratic fields, satisfying a certain simple restriction, there is at least one which satisfies the above version of Artin's conjecture. Likewise, we consider an analogue of Artin's primitive root conjecture for nonunits in real quadratic fields. Given such an element, for a rational prime p which is inert in the field, the maximal order of the unit modulo  is . As before, the extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. We show that, out of any choice of 85 algebraic numbers satisfying a certain simple restriction, there is at least one which satisfies the above version of Artin's conjecture.


Gupta and Murty's method to attack the former problems raises a question regarding , where a, b are multiplicatively independent rational positive integers. It is known that there are infinitely many integers n with 'big` . We show the same property for  .