Geodesic
Diophantine equation generated by the subfield of a circular field
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 166 (2024) no. 2, pp. 147-161

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Two forms $f(x,y,z)$ and $g(x,y,z)$ of degree $3$ were constructed, with their values being the norms of numbers in the subfields of degree $3$ of the circular fields $K_{13}$ and $K_{19}$, respectively. Using the decomposition law in a circular field, Diophantine equations $f(x,y,z)=a$ and $g(x,y,z)=b$, where $a,b\in\mathbb{Z},\ a\ne0,\ b\ne 0$ were solved. The assertions that, based on the canonical decomposition of the numbers $a$ и $b$ into prime factors, make it possible to determine whether the equations $f(x,y,z)=a$ and $g(x,y,z)=b$ have solutions were proved.
Keywords: algebraic integer, norm of algebraic number, principal ideal, fundamental basis, decomposition law in circular field
Mots-clés : Galois group, Diophantine equation. 
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     title = {Diophantine equation generated by the subfield of a circular field},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {147--161},
     publisher = {mathdoc},
     volume = {166},
     number = {2},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2024_166_2_a1/}
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I. G. Galyautdinov; E. E. Lavrentyeva. Diophantine equation generated by the subfield of a circular field. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 166 (2024) no. 2, pp. 147-161. http://geodesic.mathdoc.fr/item/UZKU_2024_166_2_a1/