Unit Groups of Infinite Abelian Extensions

Abstract

Let F he a finite extensiojn field of thle rational inumbers, Q, and let AK be an infinite abelian extensioin of F. ILet S be a finite set of primie divisors of Q inicludingl the Archim-ledean one. An S-unit of X is a field element which is a local Unlit at all prime divisors of F which do oclet r-estrict oni C to a menmber of S. It is shown that the g-roup of KS-uiniits of K is the direct produLct of the group of roots of uinity of K Nxithj a free abelian grouip. The auestion of determnining the structure of the unit group in certain infinite extersions of Q arose in sOnie workL of t1he autlhor [2] on group algiebras. In that caSe, the field _K is the maximal cyclotoniic extension of F. The conclusion of the theoremn tlhat we prove has been obtained (for entirely dif-ferenit purposes) by DiBiello [i] for a restricted class of abelian ex-:tensiens, one not including the maximal cyclotomic ones. Since the result is somewhat unexpected, it seemrs worthwhile to give an exposition of it which is independent of thle applications. We first prove a simple lemma on grouips. If G is an abelian group and H a subgroup, w-e define the ptrification of H iii G, denoted (H)X0, to be the Set of all xEG suclh that xnCH for some positive integer, i. In other words, (H), is the inverse image of thie torsion subgroup of GIH under the natural map. Let p be a prime number. We define the p-pirlification of H in G, denoted (H),, to be the set of all xEG such that xP TIH for some positive integer, r. Then (H)I is the inverse image of the p-torsion subgroup of G/H-1 uncder the natural nmap. LEMIMA. Lct G be a couii tablyv generated torsion free abelian group. Assume that for any finitely generated subgroulp, Ii', there exists a subgroup, H, containing I', such that the purification of HI in G is finitely genterated. Then G is free. PROOF. Let gi, g2, . be generators for G. By induction we shall construct finitely generated subgroups, { WI 1i}, and subsets, {Bij 1 ? i}, of G such tllat for any positive integer, n, wTe have thiat gn are in TV, GITIYV is torsion free, B,, is a free basis for 17,, and B1cIB2C . cB,,. If -this can be done, then G=UilWV implies that U Bi is a free basis for C. Received by the editors November 4, 1969. AMS Sutbject Classifications. Primar-y 1065, 1066, 1250; Secondary 2080. Key W,l7ords and Phrases. Infinite field extension, abelian field extensiotn cyclotornic field extension, tinits.