Rocky Mountain Journal of Mathematics

The integral trace form of cyclic extensions of odd prime degree

Everton Luiz de Oliveira, J. Carmelo Interlando, Trajano Pires da Nóbrega Neto, and José Othon Dantas Lopes

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Abstract

Let $L/\mathbb {Q}$ be a cyclic extension of degree~$p$, where $p$ is an odd unramified prime in $L/\mathbb {Q}$. An explicit description of the integral trace form $Tr _{L/\mathbb {Q}}(x^2)|_{\mathfrak O_L}$, where~$\mathfrak O_L$ is the ring of algebraic integers of $L$, is given, and an application to finding the minima of certain algebraic lattices is presented.

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 4 (2017), 1075-1088.

Dates
First available in Project Euclid: 6 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1501984940

Digital Object Identifier
doi:10.1216/RMJ-2017-47-4-1075

Mathematical Reviews number (MathSciNet)
MR3689945

Zentralblatt MATH identifier
06790005

Subjects
Primary: 11E12: Quadratic forms over global rings and fields 11H31: Lattice packing and covering [See also 05B40, 52C15, 52C17] 11H50: Minima of forms 11R18: Cyclotomic extensions 11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10]

Keywords
Cyclotomic fields cyclic extensions Galois module theory integral trace forms lattice packings

Citation

Oliveira, Everton Luiz de; Interlando, J. Carmelo; Neto, Trajano Pires da Nóbrega; Lopes, José Othon Dantas. The integral trace form of cyclic extensions of odd prime degree. Rocky Mountain J. Math. 47 (2017), no. 4, 1075--1088. doi:10.1216/RMJ-2017-47-4-1075. https://projecteuclid.org/euclid.rmjm/1501984940


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References

  • E. Bayer, Factorisation is not unique for higher dimensional knots, Comm. Math. Helv. 55 (1980), 583–592.
  • ––––, Unimodular Hermitian and skew-Hermitian forms, J. Algebra 74 (1982), 341–373.
  • ––––, Definite Hermitian forms and the cancellation of simple knots, Arch. Math. 40 (1983), 182–185.
  • P.E. Conner and R. Perlis, A survey of trace forms of algebraic number fields, World Scientific Publishing Co., Singapore, 1984.
  • J.H. Conway and N.J.A. Sloane, Sphere packings, lattices, and groups, Springer Verlag, New York, 1999.
  • M. Epkenhans, On trace forms of algebraic number fields, Arch. Math. 60 (1993), 527–529.
  • B. Erez, The Galois structure of the trace form in extensions of odd prime degree, J. Algebra 118 (1988), 438–446.
  • J.C. Interlando, T.P. da Nóbrega Neto, T.M. Rodrigues and J.O.D. Lopes, A note on the integral trace form in cyclotomic fields, J. Alg. Appl. 14 (2015) article id 1550045.
  • H.W. Leopoldt, Uber die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. reine angew. Math. 201 (1959), 119–149.
  • G. Lettl, The ring of integers of an Abelian number field, J. reine angew. Math. 404 (1990), 162–170.
  • W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Springer Mono. Math., Springer-Verlag, Berlin, 2004.
  • E.J. Pickett and S. Vinatier, Self-dual integral normal bases and Galois module structure, Compositio Math. 149 (2013), 1175–1202.
  • P. Samuel, Algebraic theory of numbers, Hermann, Paris, 1970.
  • W. Scharlau, On trace forms of algebraic number fields, Math. Z. 196 (1987), 125–127.
  • S. Vinatier, Structure galoisienne dans les extensions faiblement ramifiées de $\mathbb{Q}$, J. Num. Theory 91 (2001), 126–152.
  • ––––, $p$-extensions faiblement ramifiées, Publ. Math. Besan., University of Franche-Comté, Besançon.