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Zbl 0906.11058
Dummit, David S.; Hayes, David R.
Checking the ${\germ p}$-adic Stark conjecture when ${\germ p}$ is archimedean. (English)
[A] Cohen, Henri (ed.), Algorithmic number theory. Second international symposium, ANTS-II, Talence, France, May 18-23, 1996. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1122, 91-97 (1996). ISBN 3-540-61581-4

The $\frak p$-adic Stark conjecture is stated in the case when $\frak p$ is archimedean. When applied to the real subfield of a cyclotomic field, it yields known information about the signs of cyclotomic units. Suppose now that $k$ is a totally real cubic field in which every norm-positive unit is totally positive. Let $H_k$ and $H^+_k$ denote the Hilbert class fields of $k$ in the wide and narrow sense. For every $k$, the field $H^+_k$ contains three subfields $K_1,K_2,K_3$, each of degree $2h$, where $h$ is the class number of $k$. Let $\varepsilon_1,\varepsilon_2,\varepsilon_3$ be the corresponding Stark units. The authors have verified that $\varepsilon_i$ is a square in $K_i$ for every $k$ of discriminant less than 150000 if $h=2$ (25 cases), and for one of the two fields with $h=6$. An explicit construction of $H_k$ and $H^+_k$ is obtained as a by-product.
[V.Ennola (Turku)]

MSC 2000:: *11Y40 Algebraic number theory computations
11R18 Cyclotomic extensions
11R16 Cubic and quartic extensions
11R37 Class field theory for global fields

Keywords: Stark conjecture; real subfield of a cyclotomic field; signs of cyclotomic units; totally real cubic field; Hilbert class fields; Stark units

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