# Independent sets in computable free groups and fields

## Russell Miller

### City University of New York

We consider maximal independent sets within various sorts of groups and fields freely generated by countably many generators. The simplest example is the free divisible abelian group, which is just an infinite-dimensional rational vector space. As one moves up to free abelian groups, free groups, and “free fields” (i.e. purely transcendental field extensions), maximal independent sets and independent generating sets both become more complicated, from the point of view of computable model theory, but sometimes in unpredictable ways, and certain questions remain open. We present the topic partly for its own sake, but also with the intention of introducing the techniques of computable model theory and illustrating some of its possible uses for an audience to which it may be unfamiliar.

This is joint work with Charles McCoy.

Russell Miller is professor of mathematics at Queens College of CUNY and also at the CUNY Graduate Center. He conducts research in mathematical logic, especially computability theory and its interaction with other areas of mathematics, as in computable model theory. He received his doctorate from the University of Chicago in 2000, as a student of Robert Soare, and subsequently held a postdoctoral position at Cornell University until 2003, when he came to CUNY.