The multiplicative rationals

D.K Harrison
1987 Journal of Algebra  
We are interested in whether the rational numbers, or a number field in general, can be characterized purely multiplicatively. We give such a characterization, of the rational numbers only, but we use an action of S, which is new. We are also interested in characters of the rational numbers; we give an axiomatic treatment of them. We write N* for the set of positive integers, N for the set of nonnegative integers. We write 77, for { 0, 1 }. We will work with four levels of complexity: a
more » ... " an "augmented system," a "covered augmented system," and a "bicovered augmented system." By a system we mean a tuple A = ((A, e), R K W, where A is an abelian group, e is an element in A of order 2, R is a finite set of homomorphisms from A to Z" each taking P to 1, V is an infinite set of surjective homomorphisms from A onto B, and W is a map from R to the set of subsets of A, such that four axioms hold. For the second axiom we need the group Y (or more precisely, Y(A)) which is the direct sum (i.e., coproduct) of: copies of Z" one for each u E R, and copies of Z, one for each UE V. The four axioms are: (sl) UE A, u(a) =0 for all but finitely many UE V; (~2) the natural map 4: A + Y, a + (..., u(a),...), has finitely generated kernel and finite cokernel; (~3) S a finite subset of Ru V, UE V, u$S imply 3a~A with u(a)=O, VuESand u(a)=l; and 40
doi:10.1016/0021-8693(87)90179-7 fatcat:afujleg6rjd2jhwuxkw6zt73gq