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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 $F \in ℤ [x, z]$ 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 class number congruence for cyclotomic fields and their subfields

Tauno Metsänkylä (1973)

Acta Arithmetica

A complete generalization of Yokoi's p-invariants

R. Mollin, H. Williams (1992)

Colloquium Mathematicae

A correction to the paper "Upper bounds for class numbers of real quadratic fields" (Acta Arith. 68 (1994), 141-144)

Maohua Le (1995)

Acta Arithmetica

A determinant formula for congruence zeta functions of maximal real cyclotomic function fields

Daisuke Shiomi (2009)

Acta Arithmetica

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.