Vol. 9, No. 3, 2015

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Hurwitz monodromy and full number fields

David P. Roberts and Akshay Venkatesh

Vol. 9 (2015), No. 3, 511–545

We give conditions for the monodromy group of a Hurwitz space over the configuration space of branch points to be the full alternating or symmetric group on the degree. Specializing the resulting coverings suggests the existence of many number fields with surprisingly little ramification —for example, the existence of infinitely many Am or Sm number fields unramified away from {2,3,5}.

number fields, Hurwitz spaces
Mathematical Subject Classification 2010
Primary: 14D05
Secondary: 20F36, 11R21
Received: 28 January 2014
Revised: 8 January 2015
Accepted: 18 February 2015
Published: 17 April 2015
David P. Roberts
Division of Science and Mathematics
University of Minnesota
Morris, MN 56267
United States
Akshay Venkatesh
Department of Mathematics
Building 380
Stanford University
Stanford, CA 94305
United States