[PAST EVENT] Alternate geometries of the rational numbers, an introduction to the p-adic numbers

November 9, 2012
2pm - 3pm
Jones Hall, Room 301
200 Ukrop Way
Williamsburg, VA 23185Map this location
Hensel's lemma provides a means of computing roots of certain polynomials in the integers modulo p^n for arbitrary n. The method of computing such roots has a precise analogue in computing an approximation of a root via Newton's method. To complete this analogy, a geometry is required in which approximation of an integer is equivalent to computation of the residue modulo p^n. The p-adic numbers provide just such a geometry. This talk will construct the p-adic numbers, describe their geometric structure and examine some computations that their geometry affords.

This talk will assume a casual familiarity with the construction of the real numbers as a topological completion of the rational numbers, an introductory understanding of integer arithmetic modulo p^n and the use of base-p or p-ary notation in representing the natural numbers.