# Dihedral extensions and the Ankeny - Artin - Chowla conjecture

Jensen and Yui (Polynomials with Dp as Galois group J. Number Theory 15, 347-375 (1982)) proved that if p = 4n+1 is a regular prime, then there is no normal extension of the rationals with Galois group Dp (dihedral of order 2p) ramified only at p. When I first read it I noticed that such an extension exists if and only if p divides u, where $t+u\sqrt{p}$ is the fundamental unit of the real quadratic number field with discriminant p (Ankeny, Artin and Chowla conjectured that this never happens; it is known that this property is equivalent to the divisibility of the Bernoulli number B(p-1)/2 by p, hence implies that p is irregular).

I recall having seen this result in print a few years later, but can't find it anymore. Can anyone help me?

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reference-request tag? –  Dror Speiser Jan 28 '10 at 8:01
Maybe the result is in Louboutin, Park, and Lefeuvre, Construction of the real dihedral number fields of degree $2p$, Acta Arith 89 (1999) 201-215, MR 2000g:11101. –  Gerry Myerson Mar 18 '10 at 2:02
Another paper that might be relevant is Bernat Plans, On the minimal number of ramified primes in some solvable extensions of {\bf Q}, Pac J Math 215 (2004) 381-391. –  Gerry Myerson Mar 26 '10 at 6:32