Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

The construction of unramified cyclic quartic extensions of $ Q(\sqrt m)$


Author: Theresa P. Vaughan
Journal: Math. Comp. 45 (1985), 233-242
MSC: Primary 11R11; Secondary 11R29
DOI: https://doi.org/10.1090/S0025-5718-1985-0790656-6
MathSciNet review: 790656
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give an elementary general method for constructing fields K satisfying $ [K:Q] = 8$, the Galois group of K over Q is dihedral, and K is unramified over one of its quadratic subfields. Given an integer m, we describe all such fields K which contain $ Q(\sqrt m )$. The description is specific and is given in terms of the arithmetic of the quadratic subfields of K.


References [Enhancements On Off] (What's this?)

  • [1] Harvey Cohn, "Cyclic-sixteen class fields for $ Q{( - p)^{1/2}}$ by modular arithmetic," Math. Comp., v. 33, 1979, pp. 1307-1316. MR 537976 (80h:12007)
  • [2] Harvey Cohn, A Classical Invitation to Algebraic Numbers and Class Fields, Springer-Verlag, New York, 1978. MR 506156 (80c:12001)
  • [3] Harvey Cohn, "The explicit Hilbert 2-cyclic class field for $ Q(\sqrt { - p} )$," J. Reine Angew. Math., v. 321, 1981, pp. 64-77. MR 597980 (82e:12011)
  • [4] Daniel A. Marcus, Number Fields, Springer-Verlag, New York, 1977. MR 0457396 (56:15601)
  • [5] L. Rédei & H. Reichardt, "Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkörpers," J. Reine Angew. Math., v. 170, 1934, pp. 69-74.
  • [6] A. Scholz, "Über die Beziehung der Klassenzahlen quadratischer Körper zueinander," J. Reine Angew. Math., v. 166, 1932, pp. 201-203.
  • [7] B. L. van der Waerden, Modern Algebra, Ungar, New York, 1953.
  • [8] Theresa P. Vaughan, "The discriminant of a quadratic extension of an algebraic field," Math. Comp., v. 40, 1983, pp. 685-707. MR 689482 (84e:12006)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11R11, 11R29

Retrieve articles in all journals with MSC: 11R11, 11R29


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1985-0790656-6
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society