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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 class number criterion for the equation $(x^{p} - 1) / (x - 1) = p y^{q}$

Benjamin Dupuy (2007)

Acta Arithmetica

A class of diophantine equations connected with certain elliptic curves over $Q (\sqrt{- 13})$

R. J. Stroeker (1979)

Compositio Mathematica

A Class of Discrete Spectra of Non-Pisot Numbers

Dragan Stankov (2008)

Publications de l'Institut Mathématique

A class of irreducible polynomials

Joshua Harrington, Lenny Jones (2013)

Colloquium Mathematicae

Let $f (x) = x ⁿ + k_{n - 1} x^{n - 1} + k_{n - 2} x^{n - 2} + \dots + k ₁ x + k ₀ \in ℤ [x]$ , where $3 \leq k_{n - 1} \leq k_{n - 2} \leq \dots \leq k ₁ \leq k ₀ \leq 2 k_{n - 1} - 3$ . 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 of Periodic Jacobi-Perron Algorithms in Pure Algebraic Number Fields of Degree n...3.

Claude Levesque (1977)

Manuscripta mathematica

A class of polynomials

Andrzej Schinzel (1991)

Mathematica Slovaca

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.

A combinatorial class number formula.

J. Brzezinski (1989)

Journal für die reine und angewandte Mathematik

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