Geodesic
On certain multiplicative structures on cubic extensions
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 33, Tome 470 (2018), pp. 5-20

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Multiplicative properties of a certain correspondence between elements in cyclic cubic extension of rational number field and elements in adjusted pure cubic extensions are investigated. Shanks cubic polynomial case is considered to connect multiplication of pure cubic irrationals with a point summation on an associated elliptic curve. 
@article{ZNSL_2018_470_a0,
     author = {M. A. Antipov and K. I. Pimenov},
     title = {On certain multiplicative structures on cubic extensions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--20},
     publisher = {mathdoc},
     volume = {470},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_470_a0/}
}
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M. A. Antipov; K. I. Pimenov. On certain multiplicative structures on cubic extensions. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 33, Tome 470 (2018), pp. 5-20. http://geodesic.mathdoc.fr/item/ZNSL_2018_470_a0/