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 language | English |
|---|---|
| Pages (from-to) | 76-85 |
| Number of pages | 10 |
| Journal | Journal of Number Theory |
| Volume | 93 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2002 Jan 1 |
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All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
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A note on algebraic λc-monomials and double coverings. / Seo, Soogil.
In: Journal of Number Theory, Vol. 93, No. 1, 01.01.2002, p. 76-85.Research output: Contribution to journal › Article
TY - JOUR
T1 - A note on algebraic λc-monomials and double coverings
AU - Seo, Soogil
PY - 2002/1/1
Y1 - 2002/1/1
N2 - 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.
AB - 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.
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UR - http://www.scopus.com/inward/citedby.url?scp=0036222249&partnerID=8YFLogxK
U2 - 10.1006/jnth.2001.2714
DO - 10.1006/jnth.2001.2714
M3 - Article
AN - SCOPUS:0036222249
VL - 93
SP - 76
EP - 85
JO - Journal of Number Theory
JF - Journal of Number Theory
SN - 0022-314X
IS - 1
ER -