On Bases of Purely Cubic Fields over Quadratic Fields
Kiyoshi NAGATA · Tokyo Journal of Mathematics · 1985
Let $K/k$ be a relative algebraic number field of degree $n$ .It is known that under a certain condition there exist $n$ elements of $K$ , say $\omega_{1},$ $\cdots,$ $\omega_{n}$ , satisfyingwhere $O_{K},$ $O_{k}$ are the rings of integers of $K,$ $k$ respectively.We call a set of these $\omega_{1},$ $\cdots,$ $\omega_{n}$ a relative integral basis w.r.$t$ .$K/k$ .It is not still easy to have a relative integral basis explicitly.H. Wada have determined one in case that $k=Q(\sqrt{-3}),$ $K=k(\Psi\overline{A})$ with $A$ being an element of $k$ , in [1].In this paper, we have got a basis under some hypotheses by the same method in [1] when $k=Q(\sqrt{m}),$ $K=k(\sqrt{A})$ with $m$ being a square free rational integer and $A$ being an element of $k$ .\S 1.Now, let $m$ be a square free integer and $k$ be the field $Q(\sqrt{m})$ as we mentioned above.For some cubic free integer $A$ of $k$ , let $K$ be the field $k(\sqrt[8]{A})$ .The purpose of this paper is to get a basis $\omega_{1},$ $\omega_{2},$ $\omega_{8}$ of $O_{K}$ over $O_{k}$ , on the following hypotheses Hl, H2: Hl.Any prime ideal $\mathfrak{p}$ in $O_{k}$ which divides (3) is principal.H2. $A=fg^{2},$ $f$ and $g$ being in $O_{k}$ such that $(f)$ and $(g)$ have no square ideal factors and are relatively prime.We will see that these hypotheses Hl, H2 are sufficient for the existance of relative integral basis.But these may not be always necessary.The hypothesis H2 is necessary only for the convenience of the calculation in our method.It seems that the hypothesis Hl is more essential.But we will not discuss this problem in this paper.Put $\overline{A}=f^{2}g,$ $\theta=\Psi\overline{A}$ and $\overline{\theta}=\Psi\overline{\overline{A}}$ .By the relation $\theta^{2}=g\overline{\theta}$ , any element of $K$ can be expressed as the form $\omega=\alpha+\beta\theta+\gamma\overline{\theta}$ with $\alpha,$ $\beta,$ $\gamma ek$ .It can be easily verified that $\omega$ is in $O_{K}$ iff there exist $s,$ $t$ and $u$ in $O_{k}$ such that $3\alpha=s,$ $-3\alpha^{2}+3\beta\gamma fg=t$ and $\alpha^{8}+\beta^{8}A+\gamma^{3}\overline{A}-3\alpha\beta\gamma fg=u$ .Hence