In order to investigate the Leopoldt conjecture by an algebraic method, the author defined and studied the Fermat quotient and the level of a unit in a previous paper [cf. \textit{T. Shimada}, Tokyo Metropolitan University Mathematical Preprint Series, No. 5 (1996); see also his related paper in Acta Arith. 76, 335-358 (1996; Zbl 0867.11076)]. In this paper, he calculates the Fermat quotient of some cyclotomic units by different methods and considers its independence over the field \(F_p\) with \(p\) elements for an odd prime \(p\). For a natural \(m\) prime to \(p\), let \(F=\mathbb{Q} (\zeta_m)\), \(K=\mathbb{Q}(\zeta_{mp^n})\), \(\pi_n=\zeta_{p^n}-1\) and \(E_K\) be the unit group of \(K\). Moreover, for \(\eta\in E_K \setminus F\), let \(c(\eta)\) be the level of \(\eta\) and \(\eta\equiv x+y \pi_n^{c(\eta)}\bmod \pi_n^{c(\eta)+1}\). Then, \(\psi(\eta) =y/x \bmod p\) in \(\mathbb{Z}[\zeta_m]/(p)\) is the Fermat quotient. Furthermore, for the restriction \(\psi_r\) of \(\psi\) on the units of level \(r\), let \(\Psi_r\) be the subspace of \(\mathbb{Z}[\zeta_m]/(p)\) generated by \(\psi_r\). Then, under some assumptions, he proves \(\dim_{F_p} \psi_r\geq 1/2 \varphi(m)\).