A lower bound for the class numbers of abelian algebraic number fields with odd degree

Abstract:

Let ${\Delta _K},{h_K},{R_K}$ denote the discriminant, the class number, and the regulator of the Abelian algebraic number field $K = \mathbb {Q}(\alpha )$ with degree d, respectively. In this note we prove that if $d > 1,2\nmid d$, and the defining polynomial of $\alpha$ has exactly ${r_1}$ real zeros and ${r_2}$ pairs of complex zeros, then ${h_K} > w\sqrt {|{\Delta _K}|} /{2^{{r_1}}}{(2\pi )^{{r_2}}}33{R_K}\log 4|{\Delta _K}|$, where w is the number of roots of unity in K.