A new proof of a theorem of Kummer

Abstract

PROOF. Let (E/Q) =r+2s where r is the number of real infinite primes of E and s is the number of complex infinite primes. Thus if V is the group of units of E, V is of rank t = r+s-1. Let q be an isomorphism of E into the complex numbers. Then 4 can be extended in exactly p = (L/E) ways to L. If +(E) is real then any extension of 4 to L must also be real since p is odd. (If not, then the image of L would be of degree 2 over its maximal real subfield implying that p is even.) Thus (L/Q) =pr+2ps and L has pr real infinite primes and ps complex infinite primes. Therefore U is of rank u=pr+ps-1 =1pt+p-1. According to [1, Theorem 10.3] the Herbrand quotient of U is