Darrin Doud
322 TMCB Department of Mathematics Brigham Young University Provo, UT 84602   phone:  (801)4221204 
fax:  (801)4220504  email:  

S_{4} and S^{~}_{4} extensions of Q ramified at only one prime
Journal of Number Theory, 75 (1999), pp. 185197.
Abstract
For each prime p in a certain family of odd primes, we construct an S_{4} extension of Q unramified outside p. We show that for all p congruent to 3 modulo 8 in our family, this S_{4} extension embeds in an S^{~}_{4} extension, which is also unramified outside p. Invoking Serre's conjecture (in a proven case) allows us to relate the splitting of primes in these extensions to certain modular forms of level 1.
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Cited in
 Ian Kiming and Helena A. Verrill, On modular mod \ell Galois representations with exceptional images, J. of Number Theory 110 (2005) 236266.
 Lectures on Serre's Conjectures, in Arithmetic Algebraic Geometry, IAS/Park City Mathematics Series Volume 9, Brian Conrad and Karl Rubin, ed., 2001, pp. 143232.
 Avner Ash, Darrin Doud, and David Pollack, Galois representations with conjectural connections to arithmetic cohomology. Duke Math. J. 112 (2002), 521579.
 Arnaud Jehanne, Realization over Q of the groups \tilde A_{5} and \hat A_{5}. J. Number Theory 89 (2001), 340368.
 Nadya Markin, Galois groups with restricted ramification, Ph.D. Thesis, University of Illinois at UrbanaChampaign, 2006.
 Nadya Markin, Galois Groups with Minimal Ramification, in Proceedings of Conference on Algorithmic Number Theory 2007, Turku, 8386.
 Pilar Bayer, Computational aspects of Artin Lfunctions, in Zeta functions in Algebra and geometry, Contemporary Math. 566 (2012), 320.
 Siman Wong, Arithmetic of octahedral sextics, J. of Number Theory 145 (2014), 245272.
 Frank Calegari, Review of BuzzardGee, in Persiflage, March 3, 2015.