×

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.

MSC:

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

References:

[1] G. W. Anderson, “A double complex for computing the sign-cohomology of the universal ordinary distribution” in Recent Progress in Algebra (Taejon/Seoul, 1997) , ed. S. G. Hahn, H. C. Myung, and E. Zelmanov, Contemp. Math. 224 , Amer. Math. Soc., Providence, 1999, 1–27. · Zbl 0939.11035
[2] S. Bae, E.-U. Gekeler, P.-L. Kang, and L. Yin, Anderson’s double complex and Gamma monomials for rational function fields , preprint, 2001, http://www.math.uni-sb.de/PREPRINTS/preprint_liste.html#aktuell S. Bae, E.-U. Gekeler, and L. Yin, Distributions and \(\Gamma\)-monomials , Math. Ann. 321 (2001), 463–478. \CMP1 871 965 · Zbl 1163.11346
[3] P. Das, Algebraic gamma monomials and double coverings of cyclotomic fields , Trans. Amer. Math. Soc. 352 (2000), 3557–3594. JSTOR: · Zbl 1013.11069
[4] P. Deligne, J. S. Milne, A. Ogus, and K.-Y. Shih, Hodge Cycles, Motives, and Shimura Varieties , Lecture Notes in Math. 900 , Springer, Berlin, 1982. · Zbl 0465.00010
[5] A. Fröhlich, Central Extensions, Galois Groups, and Ideal Class Groups of Number Fields , Contemp. Math. 24 , Amer. Math. Soc., Providence, 1983. · Zbl 0519.12001
[6] N. Koblitz and A. Ogus, “Algebraicity of some products of values of the \(\Gamma\)-function,” appendix to: P. Deligne, “Valeurs de fonctions \(L\) et périodes d’intégrales” in Automorphic Forms, Representations and \(L\)-Functions (Corvallis, Ore., 1977) , Proc. Sympos. Pure Math. 33 , part 2, Amer. Math. Soc., Providence, 1979, 343–346. · Zbl 0449.10029
[7] D. S. Kubert, The universal ordinary distribution , Bull. Soc. Math. France 107 (1979), 179–202. · Zbl 0409.12021
[8] –. –. –. –., The \(\mathbbZ/2\mathbbZ\) cohomology of the universal ordinary distribution , Bull. Soc. Math. France 107 (1979), 203–224. · Zbl 0409.12022
[9] Y. Ouyang, Group cohomology of the universal ordinary distribution , J. Reine Angew. Math. 537 (2001), 1–32. · Zbl 1008.11042
[10] S. Seo, A note on algebraic \(\Gamma\)-monomials and double coverings , J. Number Theory 93 (2002), 76–85. \CMP1 892 931 · Zbl 1010.11060
[11] S. K. Sinha, Deligne’s reciprocity for function fields , J. Number Theory 63 (1997), 65–88. · Zbl 0932.11040
[12] W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field , Ann. of Math. (2) 108 (1978), 107–134. JSTOR: · Zbl 0395.12014
[13] J. Tate, Les conjectures de Stark sur les fonctions \(L\) d’Artin en \(s=0\) , Progr. Math. 47 , Birkhäuser, Boston, 1984.
[14] D. S. Thakur, Gamma functions for function fields and Drinfeld modules , Ann. of Math. (2) 134 (1991), 25–64. JSTOR: · Zbl 0734.11036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.