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

摘要

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 Q whose Galois group is 2-elementary abelian. Then we shall prove that F of degree [F : Q] greater than or equal to 8, is monogenic if and only if F = Q(root-1, root-2, root-3) = Q(zeta(24)) under a suitable condition for the case of degree 8.