Summary: In this paper, using the theory of double coverings of cyclotomic fields, we give a formula for \[ \dim_{\mathbb F_2} H^0(G,U_K/U_K^2), \] where \(K = \mathbb Q(\zeta_n)\), \(G = \text{Gal}(K/\mathbb Q)\), \(\mathbb F_2 = \mathbb Z/2\mathbb Z\) and \(U_K\) is the unit group of \(K\). We explicitly determine all the cyclotomic fields \(K = \mathbb Q(\zeta_n)\) such that \[ \dim_{\mathbb F_2} H^0(G,U_K/U_K^2)=1. \] Then we apply it to the unit square problem raised in [Y. Li and X. Zhang, J. Number Theory 128, No. 9, 2687–2694 (2008; Zbl 1236.11093)]. In particular, we prove that the unit square problem does not hold for \(\mathbb Q(\zeta_n)\) if \(n\) has more than three distinct prime factors, i.e. for each odd prime \(p\), there exists a unit, which is a square in all local fields \(\mathbb Q(\zeta_n)_{v}\) with \(v\mid p\) but not a square in \(\mathbb Q(\zeta_n)\), if \(n\) has more than three distinct prime factors.