Refined version of Hasse's Satz 45 on class number parity

For an imaginary abelian field K , Hasse [3, Satz 45] obtained a criterion for the relative class number to be odd in terms of the narrow class number of the maximal real subfield Kþ and the prime numbers which ramify in K , by using the analytic class number formula. In [4], we gave a refined version (1⁄4 ‘‘D-decomposed version’’) of Satz 45 by an algebraic method. In this paper, we give one more algebraic proof of the refined version.