Dummit and Hayes  noticed empirically that, in a number of
computational examples of Stark's conjecture, the Stark unit is a square.
Subsequently, they showed how this would follow from the truth of the
p-adic Stark conjecture.
In this paper,
we extend Dummit and Hayes' result to a much more general situation.
Let k be a totally real number field, L an abelian
extension in which the real place v ramifies, and K
the fixed field of the corresponding complex conjugation.
If the subgroup of Gal(L/k) generated by all complex
conjugations has rank strictly less than [k : Q]
(and some other very minor conditions hold), then the p-adic Stark
conjecture (for p = v) implies that the Stark unit
for K/k is a square in K .
The significance, as noted by Dummit and Hayes, is that the abelian part
of Stark's conjecture holds automatically in this case.