Wednesday, 28 March 2007

nt.number theory - 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 $t+usqrt{p}$
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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