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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 class of irreducible polynomials

Joshua Harrington, Lenny Jones (2013)

Colloquium Mathematicae

Let f ( x ) = x + k n - 1 x n - 1 + k n - 2 x n - 2 + + k x + k [ x ] , where 3 k n - 1 k n - 2 k k 2 k n - 1 - 3 . We show that f(x) and f(x²) are irreducible over ℚ. Moreover, the upper bound of 2 k n - 1 - 3 on the coefficients of f(x) is the best possible in this situation.

A class–field theoretical calculation

Cristian D. Popescu (2006)

Journal de Théorie des Nombres de Bordeaux

In this paper, we give the complete characterization of the p –torsion subgroups of certain idèle–class groups associated to characteristic p function fields. As an application, we answer a question which arose in the context of Tan’s approach [6] to an important particular case of a generalization of a conjecture of Gross [4] on special values of L –functions.

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