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Solution to a Problem of Lubelski and an Improvement of a Theorem of His

A. Schinzel (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

The paper consists of two parts, both related to problems of Lubelski, but unrelated otherwise. Theorem 1 enumerates for a = 1,2 the finitely many positive integers D such that every odd positive integer L that divides x² +Dy² for (x,y) = 1 has the property that either L or 2 a L is properly represented by x²+Dy². Theorem 2 asserts the following property of finite extensions k of ℚ : if a polynomial f ∈ k[x] for almost all prime ideals of k has modulo at least v linear factors, counting multiplicities,...

Solutions of cubic equations in quadratic fields

K. Chakraborty, Manisha V. Kulkarni (1999)

Acta Arithmetica

Let K be any quadratic field with K its ring of integers. We study the solutions of cubic equations, which represent elliptic curves defined over ℚ, in quadratic fields and prove some interesting results regarding the solutions by using elementary tools. As an application we consider the Diophantine equation r+s+t = rst = 1 in K . This Diophantine equation gives an elliptic curve defined over ℚ with finite Mordell-Weil group. Using our study of the solutions of cubic equations in quadratic fields...

Solving conics over function fields

Mark van Hoeij, John Cremona (2006)

Journal de Théorie des Nombres de Bordeaux

Let F be a field whose characteristic is not  2 and K = F ( t ) . We give a simple algorithm to find, given a , b , c ∈ K * , a nontrivial solution in  K (if it exists) to the equation a X 2 + b Y 2 + c Z 2 = 0 . The algorithm requires, in certain cases, the solution of a similar equation with coefficients in F ; hence we obtain a recursive algorithm for solving diagonal conics over ℚ ( t 1 , ⋯ , t n ) (using existing algorithms for such equations over  ℚ ) and over 𝔽 q ( t 1 , ⋯ , t n ) .

Some counter-examples in the theory of the Galois module structure of wild extensions

Stephen M. J. Wilson (1980)

Annales de l'institut Fourier

Considering the ring of integers in a number field as a Z Γ -module (where Γ is a galois group of the field), one hoped to prove useful theorems about the extension of this module to a module or a lattice over a maximal order. In this paper it is show that it could be difficult to obtain, in this way, parameters which are independent of the choice of the maximal order. Several lemmas about twisted group rings are required in the proof.

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