Monogenity for certain non-cyclic abelian extension fields
Abstract
We prove three results on monogenity of non-cyclic abelian extension fields over the rationals $\mathbb {Q}$. The first one is that for any odd prime cyclotomic field $k_p=\mathbb {Q}(\zeta _p)$ and any quadratic field $k=\mathbb {Q}(\sqrt {\ell })$ of discriminant $\ell $ prime to $p$, the non-cyclic abelian field $k_pk$ is non-monogenic if $\ell \lt -4$ with $\ell \nmid (p\pm 1)$ or $\ell \gt 4.$ The second is that if $k_n^+=\mathbb {Q}(\zeta _n+\zeta _n^{-1})$ is the maximal real subfield of an $n$th cyclotomic field with $n\ge 5$, $n\not \equiv 2 \pmod 4$ and if $k=\mathbb {Q}(\sqrt{-m})$ is an imaginary quadratic field of discriminant $-m \lt -4$ with $(m, n)=1,$ then the composite field $k_n^+k$ is non-monogenic. The third is that a maximal imaginary subfield $\mathbb {Q}(\zeta _q-\zeta ^{-1}_{q})$ of a cyclotomic field $k_{q}$ with $q \gt 4$, $q\equiv 0\pmod 4$ is monogenic.
