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

TL;DR

This paper characterizes when quadratic polynomials are irreducible in the ring of formal power series over integers, linking their reducibility to properties over p-adic integers.

Contribution

It provides necessary and sufficient conditions for irreducibility of quadratic polynomials in Z[[x]], connecting it to reducibility over Z_p[x].

Findings

Quadratic polynomials are reducible in Z[[x]] if and only if reducible over Z_p[x].

Established criteria for irreducibility in formal power series ring.

Linked reducibility in formal power series to p-adic polynomial reducibility.

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.