Kronecker-Weber plus epsilon. (English) Zbl 1056.11060

Let \(\overline{\mathbb{Q}}\) be the algebraic closure of the field \(\mathbb{Q}\) of rational numbers in the field \(\mathbb{C}\) of complex numbers, \(G= \text{Gal}(\mathbb{Q}/\mathbb{Q})\), \(G^{ab}\) the abelianization of \(G\), and \(\mathbb{Q}^{ab}/\mathbb{Q}\) the corresponding Galois extension. Then \(\mathbb{Q}^{ab}\) is the compositum of all abelian subextensions of \(\overline{\mathbb{Q}}/\mathbb{Q}\), and an explicit description by roots of unity is given by the well known Kronecker-Weber theorem. Let \(\mathbb{Q}^{{ab}+\varepsilon}\) be the compositum of all subfields of \(\overline{\mathbb{Q}}\) that are quadratic over \(\mathbb{Q}^{ab}\) and Galois over \(\mathbb{Q}\). A. Fröhlich [Central extension, Galois groups, and ideal class groups of number fields, Contemp. Math. 24, Am. Math. Soc., Providence (1983; Zbl 0519.12001)] determined the structure of \(\text{Gal}(\mathbb{Q}^{ab+ \varepsilon}/\mathbb{Q})\). The purpose of the present paper is to give an explicit description of \(\mathbb{Q}^{ab+\varepsilon}\).
Let \({\mathcal A}\) be the free abelian group generated by the classes \([a]\in\mathbb{Q}/\mathbb{Z}\). Let \(\sin: {\mathcal A}\to\mathbb{Q}^{ab\times}\) be the unique homomorphism such that: \(\sin[0]= 1\), and \(\sin[a]= 2\sin\pi a\), for all \(a\in\mathbb{Q}\), \(0< a< 1\). For all prime numbers \(p\), \(q\), such that \(p< q\), the author defines elements \(a_{pq}\in{\mathcal A}\), and establishes the equality: \(\mathbb{Q}^{ab+\varepsilon}= \overline{\mathbb{Q}}(\{\root 4\of{l}\), \(l\) prime}, \(\sqrt{\sin a_{pq}})\).
The paper originates from [P. Das, Algebraic \(\Gamma\)-monomials and double coverings of cyclotomic fields, Trans. Am. Math. Soc. 352, 3557–3594 (2000; Zbl 1013.11069)], where properties of \(\sin a_{pq}\) were investigated, and [S. Seo, A note on algebraic \(\Gamma\)-monomials and double coverings, J. Number Theory 93, No. 1, 76–85 (2002; Zbl 1010.11060)], where a relation between Legendre symbol and the \(p\)-valuation of \(\sin a_{pq}\) was proven.


11R20 Other abelian and metabelian extensions
11R32 Galois theory
11R34 Galois cohomology
11R37 Class field theory
Full Text: DOI arXiv


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