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 $K^+$ and the prime numbers which ramify in $K$, by using the analytic class number formula. In , we gave a refined version (= "$\Delta$-decomposed version") of Satz 45 by an algebraic method. In this paper, we give one more algebraic proof of the refined version.
"Refined version of Hasse's Satz 45 on class number parity." Tsukuba J. Math. 38 (2) 189 - 199, March 2015. https://doi.org/10.21099/tkbjm/1429103720