DOI QR코드

DOI QR Code

A GALOIS EXTENSION WITH GALOIS GROUP DIHEDRAL GROUP OR GENERALIZED QUATERNION GROUP

  • Published : 2005.10.01

Abstract

Let L/F be a Galois quadratic extension such that F contains a primitive n-th root of 1. Let N = L(${\alpha}^{{\frac{1}{n}}$) where ${\alpha}{\in}L{\ast}$. We show that if $N_{L/F}({\alpha})\;{\in}L^n{\cap}F$, and [N : L] = m, then $G(N/ F) {\simeq}D_m$ or generalized quaternion group whether $N_{L/F}({\alpha})\;{\in}\;F^n\;or\;{\notin}F^n$, respectively.

Keywords

References

  1. Y.-S. Hwang, The corestrictions of valued division algebras of Henselian Fields I, Pacific J. Math. 170 (1995), 53-81 https://doi.org/10.2140/pjm.1995.170.53
  2. N. Jacobson, Basic Algebra II, Freeman and Company, 1980
  3. G. Maile and B. H. Matzat, Inverse Galois Theory, Springer, Berlin, 1999
  4. M. Suzuki, Group Theory I, Springer-Verlag, New York, 1982
  5. J.-P. Tignol, On the corestriction of central simple algebras, Math. Z. 194 (1987), 267-274 https://doi.org/10.1007/BF01161974
  6. H. Volklein, Groups as Galois Groups: An Introduction, Cambridge Univ. Press, Cambridge, England, 1996