For more information about this meeting, contact Robert Vaughan.

Title: | Divisibility of order and first order definability in algebraic extensions of global fields |

Seminar: | Algebra and Number Theory Seminar |

Speaker: | Alexandra Shlapentokh, East Carolina University |

Abstract: |

Let M be a global field (i.e. a number field or a function field over a finite field of constants) or an infinite algebraic extension of a global field. Let O_M be the ring of integers of M in the case of characteristic equal to 0, and let O_M be the ring of integral functions of M in the case of positive characteristic. We investigate the first-order arithmetic definability of O_M over M and big subrings of M. In particular we are interested in the existence of definitions of the form
O_M = {t\in M | E_1x_1...E_kx_kP(t,x_1,...,x_k)}
where E_i is either a universal or an existential quantifier,
P(t,x_1,...,x_k) \in M[t,x_1,...,x_k],
and the number of universal quantifiers is as small as possible. We are also interested in making these definitions uniform across various classes of fields. In our talk we review the results of J. Robinson, Rumely, Poonen, Koenigsmann and Videla, and discuss a definition of divisibility of order uniform across global fields and leading to a definition of algebraic integers using two universal quantifiers uniformly across number fields (duplicating results of Poonen), and to a two-universal quantifier definition of integral functions uniform across classes of global fields (depending on the characteristic). We also produce a non-uniform definition of polynomial rings and rings of integral functions over “almost” any global function field using just one universal quantifier. Finally we discuss using divisibility of order to produce definitions of integers and integral functions in infinite algebraic extensions of global fields. This is a report on work in progress. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 12 / 15 / 2011 |

Time: | 11:15am - 12:05pm |