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Wednesday, October 8, 2014

**Abstract:** The extensions of the rational numbers obtained by adjoining a root of unity are called cyclotomic extensions. Every abelian extension of the rational numbers is contained in a cyclotomic extension. Since the field of rational functions over a finite field shares many properties with the rational numbers, it is natural to ask whether a similar phenomenon happens in this case. Carlitz (1938) found a class of function field extensions with properties strikingly similar to the cyclotomic extensions. Indeed, these extensions turn out to contain all the abelian extensions of the rational function field. In our talk we will focus on highlighting the similarities between these fields and how the geometry of the function fields often simplifies results imported from the rational numbers.