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Darrin Doud

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

phone: (801)422-1204 fax: (801)422-0504 e-mail: doud@math.byu.edu

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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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