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Darrin Doud

214 TMCB Department of Mathematics Brigham Young University Provo, UT 84602

phone: (801)422-1204 fax: (801)422-0504 e-mail:

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S₄ and S^~₄ extensions of Q ramified at only one prime

Journal of Number Theory, 75 (1999), pp. 185-197.

Abstract For each prime p in a certain family of odd primes, we construct an S₄ extension of Q unramified outside p. We show that for all p congruent to 3 modulo 8 in our family, this S₄ 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) 236--266.
• Lectures on Serre's Conjectures, in Arithmetic Algebraic Geometry, IAS/Park City Mathematics Series Volume 9, Brian Conrad and Karl Rubin, ed., 2001, pp. 143-232.
• Avner Ash, Darrin Doud, and David Pollack, Galois representations with conjectural connections to arithmetic cohomology. Duke Math. J. 112 (2002), 521--579.
• Arnaud Jehanne, Realization over Q of the groups \tilde A₅ and \hat A₅. J. Number Theory 89 (2001), 340-368.
• Nadya Markin, Galois groups with restricted ramification, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2006.
• Nadya Markin, Galois Groups with Minimal Ramification, in Proceedings of Conference on Algorithmic Number Theory 2007, Turku, 83-86.
• Pilar Bayer, Computational aspects of Artin L-functions, in Zeta functions in Algebra and geometry, Contemporary Math. 566 (2012), 3-20.
• Siman Wong, Arithmetic of octahedral sextics, J. of Number Theory 145 (2014), 245-272.
• Frank Calegari, Review of Buzzard-Gee, in Persiflage, March 3, 2015.