Remarks on principal factors in a relative cubic field

Determination of principal factors in 𝒬(√𝒟) and 𝒬(\root3\of𝒟)

“…Using the results of [2] together with a later result of Halter-Koch [5], we get the following Theorem. Consider the equation (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) NL/a(E) = p.…”

“…Let p be a primitive cube root of unity and put Ü = 2(p), L = K(p). If H is the class number of L and h is the class number of K, then (1.12) H = rh2/3, where r = 1 or 3, see [2]. Using the results of [2] together with a later result of Halter-Koch [5], we get the following Theorem.…”

“…However, we cannot distinguish between (ii) and (iii). In [2] (ii) holds. We do note that if D is a prime of the form 9r -1, then (iii) holds.…”

“…where (t, v) = (0, 0), (1,2) or (2, 1) is a principal factor whenever N(a) is. Let p be a primitive cube root of unity and put Ü = 2(p), L = K(p).…”

The Simplest Cubic Fields

“…From (2), P = 1 (mod 3), and for all q =£ P, the polynomial x3 -ax2 -{a + 3)x -1 splits completely (mod q) or is irreducible (mod q) according as On the other hand, (12) lim fJC(s)/f(s) = ^.…”