The unit equation over cyclic number fields of prime degree

On the Distribution of Cyclic Number Fields of Prime Degree

Let N Cp (X) denote the number of C p Galois extensions of Q with absolute discriminant ≤ X. A well-known theorem of Wright [1] implies that when p is prime, we have N Cp (X) = c(p)X 1 p−1 + O(X 1 p) for some positive real c(p). In this paper, we improve this result by reducing the secondary error term to O(X

Solvability of norm equations over cyclic number fields of prime degree

Let L = Q[α] be an abelian number field of prime degree q, and let a be a nonzero rational number. We describe an algorithm which takes as input a and the minimal polynomial of α over Q, and determines if a is a norm of an element of L. We show that, if we ignore the time needed to obtain a complete factorization of a and a complete factorization of the discriminant of α, then the algorithm run...

Norm-euclidean Cyclic Fields of Prime Degree

Let K be a cyclic number field of prime degree `. Heilbronn showed that for a given ` there are only finitely many such fields that are normEuclidean. In the case of ` = 2 all such norm-Euclidean fields have been identified, but for ` 6= 2, little else is known. We give the first upper bounds on the discriminants of such fields when ` > 2. Our methods lead to a simple algorithm which allows one...

Finding Normal Integral Bases of Cyclic Number Fields of Prime Degree

On the number of elliptic curves with prescribed isogeny or torsion group over number fields of prime degree

Let p be a prime and K a number field of degree p. We count the number of elliptic curves, up to K-isomorphism, having a prescribed property, where this property is either that the curve contains a fixed torsion group as a subgroup, or that it has an isogeny of prescribed degree. We also study the following question: for a given n such that |Y0(n)(Q)| > 0, does every elliptic curve over K with ...