Event Type:

Department Colloquium

Date/Time:

Friday, November 17, 2006 - 07:00

Location:

Kidd 364

Guest Speaker:

Institution:

Reed College

Abstract:

In the 1970s, John Horton Conway introduced the surreal numbers , a number system that contains not only the real numbers, but also contains many infinite numbers as well. Among the elements of this field, one finds ω (the first infinite ordinal), ω - 1 , &radic ω , 1/ω (an infinitesimal) and as well as numbers like ω^ω/&radic(ω - &pi) +ω^(1/ω). Within the surreal numbers, there is also a natural generalization of the integers, called the omnific integers, denoted OZ. The first three numbers listed above are omnific integers, while the last two are not. With this new notion of integer, one can revisit many of the classical questions of number theory. Most immediately, what are the prime numbers? Can every omnific integer be factored uniquely into omnific prime numbers? A structure theorem tells us that every surreal number can be represented as a certain generalized power series, so these questions are related to algebraic properties of generalized power series rings. In this talk, after introducing the surreal numbers, we'll explore some questions about primes in these rings.