A CHARACTERIZATION OF ALGEBRAIC NUMBER FIELDS WITH CLASS NUMBER TWO

depends only on a. Suppose first that h = 2 and consider the factorization into prime ideals (2) (a) = pi ■ • • psti • • « Xt, where the py are principal ideals while the ry are not. Then Pi = (O (j =■= 1, • ■ ■ , s). Since h = 2, it follows that r¿ry = (pa) (i,j = 1, • • • , t); moreover t must be even, =2u, say. Thus every factorization into primes implied by (2), for example Received by the editors August 3, 1959. 1 Research sponsored by National Science Foundation grant NSF G-9425.