Aurifeullian factorization and a deformation of dirichlet's class number formula

Abstract

For odd square-free $n > 1$, the cyclotomic polynomial $\Phi_n(x)$ satisfies the following identities, $$4 \Phi_n(x) = A_n(x)^2 - (-1)^{\frac{n-1}{2}} nB_n(x)^2$$, $$ \Phi_n ((-1)^{\frac{n-1}{2}}x) = C_n(x)^2-nxD_n(x)^2$$, where $A_n(x),\; B_n(x),\; C_n(x),\; D_n(x) \in Z[x]$. In this paper, we construct some units in $Z[\zeta_n]^{\ast}$ using them. Furthermore, we give a deformation of class number formula through the polynomials that appear in Aurifeullian factorization.