Diophantine Equation Generated by the Subﬁeld of a Circular Field - I. G. Galyautdinov; E. E. Lavrentyeva (Учёные записки Казанского университета: Серия Физико-математические науки)
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 K13 and K19 , 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 ∈ Z, a 6= 0, b 6= 0 were solved. The assertions that, based on the canonical decomposition of the numbers a and 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.