On the monogenity of cyclic quartic elds of prime conductor

Abstract

The characterization of an algebraic number eld K whether the ring ZK of integers of K has a power integral basis or not, is known as Hasse's problem. In nitely many real 2-elementary abelian elds L with degree 8 are proved to be non-monogenic by solving seven linear Diophantine equations associated with seven quadratic sub elds of octic eld L except for the 24th cyclotomic eld Q(e2 i=24): Based on the open problem 6 in W. Narkiewich's book related to the existence of a power integral basis, we want to give a new simple proof for non-monogenity of any cyclic quartic eld K of prime conductor p congruent 1 modulo 4 over the rationals Q except for the 5th cyclotomic eld Q(e2 i=5): This phenomenon was rst proved by using the Gauss sum attached to a quartic character by Nakahara. Our emphasis is to apply a single Diophantine equation associated with the quadratic sub eld of K to give a totally di erent and succinct proof from the previous one for non-monogenity of cyclic quartic elds K: Next we determine the monogenity of imaginary, and real biquadratic elds K over the eld Q of rational numbers and the relative monogenity of K over its quadratic sub eld k. An integral basis of the eld K is determined. For the succinct proof, the relative norm with respect to K=k of partial di erents 􀀀 of the di erent d( ) of an integer is determined and a single linear Diophantine equation consisted of three relative norms of the partial di erents with unit coe cients is considered. The idea is then extended to characterize an octic biquartic eld. We characterize the monogenity of non-cyclic but abelian octic number eld L over the rationals Q; which is composed by a cyclic quartic eld of an odd prime conductor ` and a quadratic eld (of prime conductor jp j) with prime discriminant p : In the case of odd conductor `jp j; the linear Diophantine equation with unit coe cients in a speci ed quadratic sub eld of L is applied to determine the monogenity of the octic eld L: The cases of even conductors, 5jp j; with p = 􀀀22;􀀀23 and 23; for seven octic elds L of conductors 5j2 j; it is shown that the unknown four maximal imaginary sub elds L of the 40th cyclotomic eld are non-monogenic. The nonmonogenity is proved by evaluating the absolute norm of a partial factor 􀀀 of the di erent d( ) = Q 2G(L=Q)nf g( 􀀀 ) with a suitable Galois action in the Galois group G(L=Q) for any integer in L: