Non-cyclic Algebras with Pure Maximal Subfields*

One of the most elementary consequences of the assumption that a normal division algebra A is cyclic of degree n over its centrum K is that A contains a quantity j whose minimum equation is co=7 in K. In 1933 I conjectured the truth of the converse proposition. The proof is easily reducible f to the case where n is a power p of a prime p. Let a be the characteristic of K. I succeeded in proving the theorem for q = p, e arbitrary, J as well as for a^p, e = l.§ There remained the case q^p, e^2. My hope for the truth of the theorem was heightened by H. Hasse's remark|| that it would provide an essential simplification of the arithmetic existence properties required for the proof of the theorem that all normal division algebras over an algebraic number field are cyclic. However this hope is at an end. For the conjecture is actually false in the remaining case. This is shown by a demonstration of the validity of the following theorem :