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Stark’s conjectures and Hilbert’s twelfth problem. (English) Zbl 0986.11074

The author discusses the explicit construction of real (relative) abelian extensions via Stark units. In the first section, an appropriate version of Stark’s conjecture is introduced. It is followed by a constructive proof of a known conditional theorem stating that under the assumption of Stark’s conjecture the maximal real abelian extension of a number field \(K\) can be generated by Stark units. This method is then used for actually computing real abelian extensions. Two examples illustrate the ideas, the calculation of the Hilbert class field of the real quadratic field of discriminant 328 (absolute degree 8), and of a ray class field of the totally real cubic field of discriminant 229, the module being a prime ideal of degree one above 37 (absolute degree 9).
Reviewer: M.Pohst (Berlin)

MSC:

11Y40 Algebraic number theory computations
11R20 Other abelian and metabelian extensions
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