A note on algebraic \(\Gamma\)-monomials and double coverings
We discuss algebraic \(\Gamma\)-monomials of Deligne. Deligne used the theory of Hodge cycles to show that algebraic \(\Gamma\)-monomials generate Kummer extensions of certain cyclotomic fields. \textit{P. Das} [Trans. Am. Math. Soc. 352, 3557-3594 (2000; Zbl 1013.11069)], using a double complex of Anderson and Deligne's results, showed that certain powers of algebraic \(\Gamma\)-monomials and certain square roots of sine monomials generate abelian extensions of \(\mathbb{Q}\). Das also gave one example of a nonabelian double covering of a cyclotomic field generated by the square root of a sine monomial. In this note, we will produce infinitely many examples of nonabelian double coverings of cyclotomic fields of Das type. The construction of the examples depends in an interesting way on a lemma of Gauss figuring in an elementary proof of quadratic reciprocity.
scientific article; zbMATH DE number 1753719