1. The basics of algebraic number theory. An algebraic number field is a field K = Q(a) where a is a zero of an irreducible (over Q) polynomial f(x) with integral coefficients. The degree of K, which we denote by n = n(K) = [K:Q], is the degree of ƒ(%). We write the roots of /(x) = 0 as a, a, • • • ,a ( n ) m such a way that for l^j^r1 = r1(K), a Q) is real, while for j>ru a 0 ) is complex. If we let n = ri+2r2, then it is customary to order the rr=r2(K) complex conjugate pairs of roots so that for r i+ l ^ j ^ r i+ r 2 , a=a The a ( , ) s are called the conjugates of a and the fields K ( , =Q(a) are called the conjugate fields of K. If r2 = 0, we say K is totally real and if ri=0, we say K is totally complex. The integers of K are those elements of K which are zeros of a polynomial with integer coefficients and leading coefficient 1. The integers of K form a ring which we denote by o. As is well known, factorization of the integers of K into prime integers is not necessarily unique. Various equivalent ways of remedying this have been used; we follow Dedekind's method. If a i , • • • , ak are elements of X, the set