A note on algebraic λc-monomials and double coverings

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Abstract

We discuss algebraic Γ-monomials of Deligne. Deligne used the theory of Hodge Cycles to show that algebraic Γ-monomials generate Kummer extensions of certain cyclotomic fields. Das, using a double complex of Anderson and Deligne's results, showed that certain powers of algebraic Γ-monomials and certain square roots of sine monomials generate abelian extensions of 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 non-abelian 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.

Original languageEnglish
Pages (from-to)76-85
Number of pages10
JournalJournal of Number Theory
Volume93
Issue number1
DOIs
Publication statusPublished - 2002

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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