Geodesic
Galois groups associated with CM-fields, skew-symmetric matrices and orthogonal matrices
Glasgow mathematical journal, Tome 32 (1990) no. 1, pp. 35-46

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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. 
Cohen, S. D.; Odoni, R. W. K. Galois groups associated with CM-fields, skew-symmetric matrices and orthogonal matrices. Glasgow mathematical journal, Tome 32 (1990) no. 1, pp. 35-46. doi: 10.1017/S0017089500009046
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