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Galois groups associated with CM-fields, skew-symmetric matrices and orthogonal matrices

Published online by Cambridge University Press:  18 May 2009

S. D. Cohen  and
R. W. K. Odoni
S. D. Cohen
Affiliation:
Department of MathematicsUniversity of Glasgow GlasgowG12 8QW
R. W. K. Odoni
Affiliation:
Department of MathematicsUniversity of Exeter ExeterEx4 4QJ
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The main aims of this paper are to provide a device for constructing large families of complex-multiplication (CM) fields, and to examine the Galois groups of some related field extensions. We recall that an algebraic number field K (i.e. ) is called a CM-field if it is totally complex but is quadratic over some totally real field (see Section 1). CM fields are important in algebraic geomety, since the ring of endomorphisms of a simple abelian variety defined over an algebraic number field is either ℤ or a ℤ-order in a CM field. Moreover, CM fields figure prominently in classfield theory, since Shimura [15] has shown that “almost all” classfields over CM fields K can be generated by means of division points on abelian varieties admitting ℤ-orders in K as their endomorphism rings. Shimura's work can be regarded as a natural generalization of the classical method (due to Kronecker and H. Weber) of generating classfields of imaginary quadratic fields via division points on CM-elliptic curves.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 32 , Issue 1 , January 1990 , pp. 35 - 46
Copyright
Copyright © Glasgow Mathematical Journal Trust 1990

References

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