Note on the ideals of cyclic algebras
Bulletin of the American Mathematical Society
1. Introduction. The purpose of this note is the generalization of certain results in a recent paper by Latimer f on the number of ideals of given norm in a generalized quaternion algebra. We consider rational cyclic division algebras D of prime degree n(*z2) over the field R of rational numbers. Let o be any maximal order J of D. The reduced discriminant of o is an invariant A=A(D) of D called the discriminant of D, and is of the form A= ±cr w(n-1) , where a = qi • • • q s is the product of
... distinct rational primes q\ • • • q s which are ramified § in D. For each such g, the two-sided ideal qo is the nth power of an indecomposable two-sided prime ideal of o, and the g-adic extension D q is a division algebra of degree n of R q . For all other rational primes p, D p is the algebra of all matrices of degree n over R p and op is a two-sided prime ideal of o having only one-sided ideal divisors. By a (normal) ideal of D is meant an ideal (one or two-sided) with respect to some maximal order of D. Such an ideal is called integral if it is contained in its right or left order. We denote various maximal orders by o, Oi, 02, • • • , and an ideal a by a»,-if o»a = aOj = a and it is necessary to indicate 0; and Oy. The (reduced) norm of an ideal is an ideal of R such that, for a principal ideal ao (or oa), a in D, N(ao) (or N(oa)) = (N(a)), where N(a) is the reduced norm corresponding to the rank equation of degree n. Our object is to prove the following result.