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Ideal arithmetic and infrastructure in purely cubic function fields

Renate Scheidler (2001)

Journal de théorie des nombres de Bordeaux

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,...

Ideal class groups, Hilbert’s irreducibility theorem, and integral points of bounded degree on curves

Aaron Levin (2007)

Journal de Théorie des Nombres de Bordeaux

We study the problem of constructing and enumerating, for any integers $m, n > 1$ , number fields of degree $n$ whose ideal class groups have “large" $m$ -rank. Our technique relies fundamentally on Hilbert’s irreducibility theorem and results on integral points of bounded degree on curves.

Ideal Class Groups in Basic ...p1 x....x...ps-Extensions of Abelian Number Fields.

Eduardo C. Friedmann (1981/1982)

Inventiones mathematicae

Ideal class groups of cubic cyclic fields

Shin Nakano (1986)

Acta Arithmetica

Ideal class groups of cyclotomic number fields I

Franz Lemmermeyer (1995)

Acta Arithmetica

Ideal class groups of cyclotomic number fields II

Franz Lemmermeyer (1998)

Acta Arithmetica

Ideal class groups of cyclotomic number fields III

Franz Lemmermeyer (2008)

Acta Arithmetica

Ideal Criteria for both Ideal Criteria for both X2-dy2 = M1 And X2-dy2 = M2 to have Primitive Solutions for any Integers M1, M2 Prime to D > 0

Mollin, R. (2002)

Serdica Mathematical Journal

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].