TY - JOUR

T1 - Cube root Ramanujan formulas and elementary Galois theory

AU - Krepkii, I. A.

AU - Pimenov, K. I.

PY - 2015/10/1

Y1 - 2015/10/1

N2 - The cube root Ramanujan formulas are explained from the point of view of Galois theory. Let F be a cyclic cubic extension of a field K. It is proved that the normal closure over K of a pure cubic extension of F contains a certain pure cubic extension of K. The proposed proof can be generalized to radicals of any prime degree q. In the case where the base field K is the field of rational numbers and the field F is embedded in the cyclotomic extension obtained by adding the pth roots of unity, the corresponding simple radical extension of the field of rational numbers is explicitly constructed. The proof of the main result illustrates Hilbert’s Theorem 90. An example of a particular formula generalizing Ramanujan’s formulas for degree 5 is given. A necessary condition for nested radical expressions of depth 2 to be contained in the normal closure of a pure cubic extension of the field F is given.

AB - The cube root Ramanujan formulas are explained from the point of view of Galois theory. Let F be a cyclic cubic extension of a field K. It is proved that the normal closure over K of a pure cubic extension of F contains a certain pure cubic extension of K. The proposed proof can be generalized to radicals of any prime degree q. In the case where the base field K is the field of rational numbers and the field F is embedded in the cyclotomic extension obtained by adding the pth roots of unity, the corresponding simple radical extension of the field of rational numbers is explicitly constructed. The proof of the main result illustrates Hilbert’s Theorem 90. An example of a particular formula generalizing Ramanujan’s formulas for degree 5 is given. A necessary condition for nested radical expressions of depth 2 to be contained in the normal closure of a pure cubic extension of the field F is given.

KW - Gaussian periods

KW - Kummer theory

KW - radical extension

KW - Ramanujan formulae

UR - http://www.scopus.com/inward/record.url?scp=84959332493&partnerID=8YFLogxK

U2 - 10.3103/S106345411504007X

DO - 10.3103/S106345411504007X

M3 - Article

AN - SCOPUS:84959332493

VL - 48

SP - 214

EP - 223

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 4

ER -