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A “class group” obstruction for the equation C y d = F ( x , z )

Denis Simon (2008)

Journal de Théorie des Nombres de Bordeaux

In this paper, we study equations of the form C y d = F ( x , z ) , where F [ x , z ] is a binary form, homogeneous of degree n , which is supposed to be primitive and irreducible, and d is any fixed integer. Using classical tools in algebraic number theory, we prove that the existence of a proper solution for this equation implies the existence of an integral ideal of given norm in some order in a number field, and also the existence of a specific relation in the class group involving this ideal. In some cases, this result...

A determinant formula for the relative class number of an imaginary abelian number field

Mikihito Hirabayashi (2014)

Communications in Mathematics

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.

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