On special units of cyclotomic number fields

Abstract

The cyclotomic units $C_p^+$ of $Q(\xi_p)^+$ are of finite index in the full unit group $E_p^+$, and $h_p^+=[E_p^+:C_p^+]$, where $h_p^+$ denotes the class number of $Q(\xi_p)^+$. Based on this fact we study how the fundamental units of real quadratic fields and simpleast cubic fields are expressed by the cyclotomic units. And we get an intresting result that for an expression of a fundamental unit, namely, $\theta=\Pi \xi^{x_a}_a$ where the signs of $x_a$ are related to Legendre``s symblol ($\frac{a}{p}$) (respectively $a^{\frac{p-1}{3}}(\bmod p)$) for quadratic cases (respectively cubic cases).