Gröbner Bases in Orders of Algebraic Number Fields

We prove that any orderOof any algebraic number field K is a reduction ring. Rather than showing the axioms for a reduction ring hold, we start from scratch by well-orderingO, defining a division algorithm, and demonstrating how to use it in a Buchberger algorithm which computes a Gröbner basis given a finite generating set for an ideal. It is shown that our theory of Gröbner bases is equivalent to the ideal membership problem and in fact, a total of eight characterizations are given for a Gröbner basis. Additional conclusions and questions for further investigation are revealed at the end of the paper.