## 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

### Citations:

Zbl 1013.11069; Zbl 1010.11060; Zbl 0519.12001
Full Text:

### References:

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