Geodesic
A determinant formula for the relative class number of an imaginary abelian number field
Communications in Mathematics, Tome 22 (2014) no. 2, pp. 133-140.

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We give a new formula for the relative class number of an imaginary abelian number field $K$ by means of determinant with elements being integers of a cyclotomic field generated by the values of an odd Dirichlet character associated to $K$. We prove it by a specialization of determinant formula of Hasse.
Classification : 11R20, 11R29
Keywords: imaginary abelian number field; relative class number; determinant; class number formula 
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     author = {Hirabayashi, Mikihito},
     title = {A determinant formula for the relative class number of an imaginary abelian number field},
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Hirabayashi, Mikihito. A determinant formula for the relative class number of an imaginary abelian number field. Communications in Mathematics, Tome 22 (2014) no. 2, pp. 133-140. http://geodesic.mathdoc.fr/item/COMIM_2014__22_2_a1/