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. Tanoé 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 $\mathbb{Q}$ whose Galois group is 2-elementary abelian. Then we shall prove that F of degree $[F:\mathbb{Q}] \geqq 8$, is monogenic if and only if $F = \mathbb{Q}(\sqrt { - 1} ,\sqrt { - 2} ,\sqrt { - 3} ) = \mathbb{Q}(\zeta _{24} )$ under a suitable condition for the case of degree 8.