Cyclic Cubic Extensions of Q

TL;DR

This paper explicitly describes all cyclic cubic extensions of Q using irreducible trinomials and constructs integral points on specific elliptic curves, revealing infinitely many primes expressible as sums of two rational cubes.

Contribution

It provides explicit descriptions of cyclic cubic extensions via trinomials and parametrizes integral points on certain elliptic curves related to these extensions.

Findings

Explicit irreducible trinomials for cyclic cubic extensions

Parametrization of integral points on specific elliptic curves

Infinitely many primes are sums of two rational cubes

Abstract

In this article we explicitly describe irreducible trinomials X^3-aX+b which gives all the cyclic cubic extensions of Q. In doing so, we construct all integral points (x,y,z) with GCD(y,z)=1, of the curves X^2+3Y^2 = 4DZ^3 and X^2+27Y^2=4DZ^3 as D varies over cube-free positive integers. We parametrise these points using well known parametrisation of integral points (x,y,z) of the curve X^2+3Y^2=4Z^3 with GCD(y,z)=1. As an accidental byproduct of our result we show that there are infinitely many primes congruent to 1 or 8 modulo 9, can be expressed as sum of two rational cubes.