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This paper investigates the arithmetic of fractional ideals of a purely cubic function field and the infrastructure of the principal ideal class when the field has unit rank one. First, we describe how irreducible polynomials decompose into prime ideals in the maximal order of the field. We go on to compute so-called canonical bases of ideals; such bases are very suitable for computation. We state algorithms for ideal multiplication and, in the case of unit rank one and characteristic at least five,...
We study the problem of constructing and enumerating, for any integers , number fields of degree whose ideal class groups have “large" -rank. Our technique relies fundamentally on Hilbert’s irreducibility theorem and results on integral points of bounded degree on curves.
This article provides necessary and sufficient conditions for
both of the Diophantine equations X^2 − DY^2 = m1 and x^2 − Dy^2 = m2
to have primitive solutions when m1 , m2 ∈ Z, and D ∈ N is not a perfect
square. This is given in terms of the ideal theory of the underlying real
quadratic order Z[√D].
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