Chatelain’s integral bases for triquadratic number fields

Abstract

Let $K_{n}=\mathbb {Q}( \alpha _{2^{0}},\ldots ,\alpha _{2^{n-3}},~\mu ~\alpha _{2^{n-2}},~\mu ^{\prime }\alpha _{2^{n-1}})$ be the compositum of the family $(\mathbb {Q} ( \alpha _{2^{k}}) ) _{0\le k\le n-1}$ of n quadratic fields which are distinct in pairs. $K_{n}$ is so called n-quadratic number field. In our paper, using Danielle Chatelain’s method, we calculate an integral basis of the integral ring $\mathbb {Z}_{K_{3}}$ of the triquadratic field $K_{3}= \mathbb {Q} ( \sqrt{dm},\sqrt{dn},\sqrt{d^{\prime }m^{\prime }n^{\prime }l}) .$ We give both an integral basis $\mathfrak {B}_{K_{3}}$ $\left( i.e\text { a }\mathfrak { \mathbb {Z} }\text {-basis of } \mathbb {Z} _{K_{3}}\right)$ called the $\mathfrak { \mathbb {Z} }\text {-}basis$ of Chatelain for $\mathbb {Z} _{K_{3}}$ and the discriminant $\mathfrak {D}_{K_{3}/ \mathbb {Q} }$ of $K_{3}$ over $\mathbb {Q}$ in the following complete and general cases: $( dm,dn,d^{\prime }m^{\prime }n^{\prime }l) \equiv \left( 1,1,1\right) ,\left( 1,1,2\right) ,\left( 1,1,3\right)$ and $\left( 1,2,3\right)$ $(\mathrm {mod} \ 4),$ where the square-free integers dm,  dn and $d^{\prime }m^{\prime }n^{\prime }l$ are such that: $( dm,dn) =d,$ $\left( dmn,l\right) =1,$ $( dm,d^{\prime }m^{\prime }n^{\prime }) =d^{\prime }m^{\prime }$ and $( dn,d^{\prime }m^{\prime }n^{\prime }) =d^{\prime }n^{\prime }.$