Generalizations of girstmair’s formulas

Abstract

In 1994 and 1995 GIRSTMAIR gave (relative) class number formulas for the imaginary quadratic field $$\mathbb{Q}(\sqrt { - p} )$$ , P an odd prime with p ≡ 3 (mod 4) and p ≥ 7, using the coefficients of the digit expression of 1/p and z/p, respectively, where z is an integer with 1 ≤ z ≤p - 1. We extend the formulas to an imaginary abelian number field.