On the representation theory for full decomposable forms

Abstract

Let k Q be any finite normal extension and fix an order D of k invariant under the galois group G( k Q ) . The ring class field K corresponding to D is normal over Q. We solve the problem of determining which full decomposable forms associated to the invertible ideals of D integrally represent a given positive integer m. First we establish a one-to-one correspondence between the improperly equivalent classes of such forms and the conjugacy classes of G( K Q ) which are contained in G( K k ) . The solution is then seen to rest upon determining the Artin class in G( K Q ) of the unramified primes dividing m. This is accomplished by evaluating certain induced characters of G( K Q ) congruentially in terms of an associated integer linear recurrence sequence.