Rational compositum genus for a pure cubic field

Algebra Seminar Torsion subgroups of rational elliptic curves over the compositum of all cubic fields

Abstract: Let E/Q be an elliptic curve and let Q(3∞) denote the compositum of all cubic extensions of Q. While the group E(3∞) is not finitely generated, one can show that its torsion subgroup is finite; this holds more generally for any Galois extension of Q that contains only finitely many roots of unity. I will describe joint work with Daniels, Lozano-Robledo, and Najman, in which we obtain ...

Torsion subgroups of rational elliptic curves over the compositum of all cubic fields

Let E/Q be an elliptic curve and let Q(3∞) be the compositum of all cubic extensions of Q. In this article we show that the torsion subgroup of E(Q(3∞)) is finite and determine 20 possibilities for its structure, along with a complete description of the Q-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures ...

Explicit decomposition of a rational prime in a cubic field

A Relative Integral Basis over Q( √−3) for the Normal Closure of a Pure Cubic Field

Let K be a pure cubic field. Let L be the normal closure of K. A relative integral basis (RIB) for L over Q(√ −3) is given. This RIB simplifies and completes the one given by Haghighi (1986).

On Rational Cubic Residues

In 1958 E. Lehmer found an explicit description of those primes p for which a given prime q is a cubic residue. Her result states that if one writes 4p = L + 27M, then q is a cubic residue if and only if M/L ≡ (t − 1)/(t − 9t) mod q for some integer t. Recently, Z. Sun has stated a similar result for cubic nonresidues which follows from several corollaries appearing in an earlier paper of his. ...