Power integral bases in algebraic number fields whose Galois groups are 2-elementary abelian

Abstract

Let K be a biquadratic field. M.-N. Gras and F. Tanoe gave a necessary and sufficient condition that K is monogenic by using a diophantine equation of degree 4 [3]. We consider algebraic extension fields of higher degree. Let F be a Galois extension field over the rationals % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFAesuaaa!40A2! $$\mathbb{Q}$$ whose Galois group is 2-elementary abelian. Then we shall prove that F of degree % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadA % eacaGG6aWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaa % cqWFAesucaGGDbGaeyyzImRaaGioaaaa!4673! $$[F:\mathbb{Q}] \geqq 8$$ , is monogenic if and only if % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2 % da9mrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8Ng % HeLaaiikamaakaaabaGaeyOeI0IaaGymaaWcbeaakiaacYcadaGcaa % qaaiabgkHiTiaaikdaaSqabaGccaGGSaWaaOaaaeaacqGHsislcaaI % ZaaaleqaaOGaaiykaiabg2da9iab-PrirjaacIcacqaH2oGEdaWgaa % WcbaGaaGOmaiaaisdaaeqaaOGaaiykaaaa!5173! $$F = \mathbb{Q}(\sqrt { - 1} ,\sqrt { - 2} ,\sqrt { - 3} ) = \mathbb{Q}(\zeta _{24} )$$ under a suitable condition for the case of degree 8.