Factorization of quadratic polynomials in the ring of formal power series over Z

Abstract

We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring $Z[[x]]$ of formal power series with integer coefficients. For $n,m\ge 1$ and $p$ prime, we show that $p^n+p^m\beta x+\alpha x^2$ is reducible in $Z[[x]]$ if and only if it is reducible in $Z_p[x]$, the ring of polynomials over the $p$-adic integers.