Ph.D Student | Joseph Cohen |
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Subject | Primitive Roots in Algebraic Number Fields |

Department | Department of Mathematics |

Supervisors | Full Professor Sonn Jack |

Full Professor Zeev Rudnick |

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 _{}.