Cubic Ramanujan formulae and elementary Galois theory

Abstract

Explanation for cubic root Ramanujan formulae is given from Galois theory perspective. Let F be a cyclic cubic extension over a base field K. It is proved that normal closure with respect to K of pure cubic extension of the field F contains certain pure cubic extension of the base field. Given proof can be generalized to the case where the radicals are of any prime degree. An explicit construction of the simple radical extension in question over a field of rational numbers is given for the field F being embedded in the cyclotomic prime field. The proof of the main result illustraes Hilbert 90 theorem. Example of Ramanujan formulae analogue for degree 5 is given. The necceccary condition for nested radical expression of depth two to belong the normal closure of the normal closure of pure cubic extension of the field F is given. Refs 5.

Keywords:

Kummer theory, Ramanujan formulae, radical extension, Gaussian periods