Tohoku Mathematical Journal

On a certain nilpotent extension over $\boldsymbol{Q}$ of degree 64 and the 4-th multiple residue symbol

Fumiya Amano

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Abstract

In this paper, we introduce the 4-th multiple residue symbol $[p_1,p_2,p_3,p_4]$ for certain four prime numbers $p_i$'s, which extends the Legendre symbol $\big(\frac{p_1}{p_2}\big)$ and the Rédei triple symbol $[p_1,p_2,p_3]$ in a natural manner. For this we construct concretely a certain nilpotent extension $K$ over $\boldsymbol{Q}$ of degree 64, where ramified prime numbers are $p_1, p_2$ and $p_3$, such that the symbol $[p_1,p_2,p_3,p_4]$ describes the decomposition law of $p_4$ in the extension $K/\boldsymbol{Q}$. We then establish the relation of our symbol $[p_1,p_2,p_3,p_4]$ and the 4-th arithmetic Milnor invariant $\mu_2(1234)$ (an arithmetic analogue of the 4-th order linking number) by showing $[p_1,p_2,p_3,p_4] = (-1)^{\mu_2(1234)}$.

Article information

Source
Tohoku Math. J. (2), Volume 66, Number 4 (2014), 501-522.

Dates
First available in Project Euclid: 21 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1432229194

Digital Object Identifier
doi:10.2748/tmj/1432229194

Mathematical Reviews number (MathSciNet)
MR3350281

Zentralblatt MATH identifier
06431044

Subjects
Primary: 11A15: Power residues, reciprocity
Secondary: 11R32: Galois theory 57M27: Invariants of knots and 3-manifolds

Keywords
Rédei triple symbol Milnor invariant 4-th multiple residue symbol

Citation

Amano, Fumiya. On a certain nilpotent extension over $\boldsymbol{Q}$ of degree 64 and the 4-th multiple residue symbol. Tohoku Math. J. (2) 66 (2014), no. 4, 501--522. doi:10.2748/tmj/1432229194. https://projecteuclid.org/euclid.tmj/1432229194


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References

  • F. Amano, On Rédei's dihedral extension and triple reciprocity law, Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), 1–5.
  • B. J. Birch, Cyclotomic fields and Kummer extensions. Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 85–93. Thompson, Washington, D.C., 1967.
  • K. T. Chen, R. H. Fox and R. C. Lyndon, Free differential calculus. IV. The quotient groups of the lower central series, Ann. of Math. (2) 68 (1958), 81–95.
  • R. H. Fox, Free differential calculus. I: Derivation in the free group ring, Ann. of Math. 57 (1953), 547–560.
  • Y. Ihara, On Galois representations arising from towers of coverings of ${\bf P}^1 \setminus \{0,1,\infty \}$, Invent. Math. 86 (1986), no. 3, 427–459.
  • H. Koch, Galois theory of $p$-extensions. With a foreword by I. R. Shafarevich. Translated from the 1970 German original by Franz Lemmermeyer. With a postscript by the author and Lemmermeyer. Springer Monographs Math. Springer-Verlag, Berlin, 2002.
  • H. Koch, On $p$-extension with given ramification, Appendix in: K. Haberland, Galois cohomology of algebraic number fields, 89–126, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.
  • J. Milnor, Link groups, Ann. of Math. 59 (1954), 177–195.
  • J. Milnor, Isotopy of links, in Algebraic Geometry and Topology, A symposium in honor of S. Lefschetz (edited by R.H. Fox, D.C. Spencer and A.W. Tucker), 280–306, Princeton University Press, Princeton, N.J., 1957.
  • M. Morishita, Milnor's link invariants attached to certain Galois groups over Q, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), 18–21.
  • M. Morishita, On certain analogies between knots and primes, J. Reine Angew. Math. 550 (2002), 141–167.
  • M. Morishita, Milnor invariants and Massey products for prime numbers, Compos. Math. 140 (2004), 69–83.
  • M. Morishita, Knots and Primes–-An introduction to arithmetic topology, Universitext, Springer, London, 2012.
  • K. Murasugi, Nilpotent coverings of links and Milnor's invariant, Low-dimensional topology (Chelwood Gate, 1982), 106–142, London Math. Soc. Lecture Note Ser., 95, Cambridge Univ. Press, Cambridge-New York, 1985.
  • T. Oda, Note on meta-abelian quotients of pro-$l$ free groups, preprint, 1985.
  • L. Rédei, Ein neues zahlentheoretisches Symbol mit Anwendungen auf die Theorie der quadratischen Zahlkörper I, J. Reine Angew. Math. 180 (1939), 1–43.
  • D. Vogel, On the Galois group of 2-extensions with restricted ramification, J. Reine Angew. Math. 581 (2005), 117–150.
  • D. Vogel, A letter to M. Morishita, 2008, February.